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RD Sharma solutions for Class 11 Mathematics chapter 23 - The straight lines

Mathematics Class 11

RD Sharma Mathematics Class 11 Chapter 23: The straight lines

Ex. 23.10Ex. 23.20Ex. 23.30Ex. 23.40Ex. 23.50Ex. 23.60Ex. 23.70Ex. 23.80Ex. 23.90Ex. 23.11Ex. 23.12Ex. 23.13Ex. 23.14Ex. 23.15Ex. 23.16Ex. 23.17Ex. 23.18Ex. 23.19Others

Chapter 23: The straight lines Exercise 23.10 solutions [Pages 12 - 14]

Ex. 23.10 | Q 1.1 | Page 12

Find the slope of the lines which make the following angle with the positive direction of x-axis:

$- \frac{\pi}{4}$

Ex. 23.10 | Q 1.2 | Page 12

Find the slope of the lines which make the following angle with the positive direction of x-axis:

$\frac{2\pi}{3}$

Ex. 23.10 | Q 1.3 | Page 12

Find the slope of the lines which make the following angle with the positive direction of x-axis:

$\frac{3\pi}{4}$

Ex. 23.10 | Q 1.4 | Page 12

Find the slope of the lines which make the following angle with the positive direction of x-axis: $\frac{\pi}{3}$

Ex. 23.10 | Q 2.1 | Page 13

Find the slope of a line passing through the following point:

(−3, 2) and (1, 4)

Ex. 23.10 | Q 2.2 | Page 13

Find the slope of a line passing through the following point:

$(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )$

Ex. 23.10 | Q 2.3 | Page 13

Find the slope of a line passing through the following point:

(3, −5), and (1, 2)

Ex. 23.10 | Q 3.1 | Page 13

State whether the two lines in each of the following are parallel, perpendicular or neither.

Through (5, 6) and (2, 3); through (9, −2) and (6, −5)

Ex. 23.10 | Q 3.2 | Page 13

State whether the two lines in each of the following are parallel, perpendicular or neither.

Through (9, 5) and (−1, 1); through (3, −5) and (8, −3)

Ex. 23.10 | Q 3.3 | Page 13

State whether the two lines in each of the following is parallel, perpendicular or neither.

Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)

Ex. 23.10 | Q 3.4 | Page 13

State whether the two lines in each of the following is parallel, perpendicular or neither.

Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).

Ex. 23.10 | Q 4 | Page 13

Find the slope of a line (i) which bisects the first quadrant angle (ii) which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.

Ex. 23.10 | Q 5.1 | Page 13

Using the method of slope, show that the following points are collinear A (4, 8), B (5, 12), C (9, 28).

Ex. 23.10 | Q 5.2 | Page 13

Using the method of slope, show that the following points are collinear A (16, − 18), B (3, −6), C (−10, 6) .

Ex. 23.10 | Q 6 | Page 13

What is the value of y so that the line through (3, y)  and (2, 7) is parallel to the line through (−1, 4) and (0, 6)?

Ex. 23.10 | Q 7.1 | Page 13

What can be said regarding a line if its slope is  zero ?

Ex. 23.10 | Q 7.2 | Page 13

What can be said regarding a line if its slope is positive ?

Ex. 23.10 | Q 7.3 | Page 13

What can be said regarding a line if its slope is negative?

Ex. 23.10 | Q 8 | Page 13

Show that the line joining (2, −3) and (−5, 1) is parallel to the line joining (7, −1) and (0, 3).

Ex. 23.10 | Q 9 | Page 13

Show that the line joining (2, −5) and (−2, 5) is perpendicular to the line joining (6, 3) and (1, 1).

Ex. 23.10 | Q 10 | Page 13

Without using Pythagoras theorem, show that the points A (0, 4), B (1, 2) and C (3, 3) are the vertices of a right angled triangle.

Ex. 23.10 | Q 11 | Page 13

Prove that the points (−4, −1), (−2, −4), (4, 0) and (2, 3) are the vertices of a rectangle.

Ex. 23.10 | Q 12 | Page 13

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: $\frac{a}{h} + \frac{b}{k} = 1$.

Ex. 23.10 | Q 13 | Page 13

The slope of a line is double of the slope of another line. If tangents of the angle between them is $\frac{1}{3}$,find the slopes of the other line.

Ex. 23.10 | Q 14 | Page 13

Consider the following population and year graph:
Find the slope of the line AB and using it, find what will be the population in the year 2010. Ex. 23.10 | Q 15 | Page 14

Without using the distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.

Ex. 23.10 | Q 16 | Page 14

Find the angle between the X-axis and the line joining the points (3, −1) and (4, −2).

Ex. 23.10 | Q 17 | Page 14

Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.

Ex. 23.10 | Q 18 | Page 14

Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.

Ex. 23.10 | Q 19 | Page 14

Find the angle between X-axis and the line joining the points (3, −1) and (4, −2).

Ex. 23.10 | Q 20 | Page 14

By using the concept of slope, show that the points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.

Ex. 23.10 | Q 21 | Page 14

A quadrilateral has vertices (4, 1), (1, 7), (−6, 0) and (−1, −9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.

Chapter 23: The straight lines Exercise 23.20 solutions [Page 17]

Ex. 23.20 | Q 1 | Page 17

Find the equation of the line parallel to x-axis and passing through (3, −5).

Ex. 23.20 | Q 2 | Page 17

Find the equation of the line perpendicular to x-axis and having intercept − 2 on x-axis.

Ex. 23.20 | Q 3 | Page 17

Find the equation of the line parallel to x-axis and having intercept − 2 on y-axis.

Ex. 23.20 | Q 4 | Page 17

Draw the lines x = − 3, x = 2, y = − 2, y = 3 and write the coordinates of the vertices of the square so formed.

Ex. 23.20 | Q 5 | Page 17

Find the equations of the straight lines which pass through (4, 3) and are respectively parallel and perpendicular to the x-axis.

Ex. 23.20 | Q 6 | Page 17

Find the equation of a line which is equidistant from the lines x = − 2 and x = 6.

Ex. 23.20 | Q 7 | Page 17

Find the equation of a line equidistant from the lines y = 10 and y = − 2.

Chapter 23: The straight lines Exercise 23.30 solutions [Page 21]

Ex. 23.30 | Q 1 | Page 21

Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.

Ex. 23.30 | Q 2.1 | Page 21

Find the equation of a straight line with slope 2 and y-intercept 3 .

Ex. 23.30 | Q 2.2 | Page 21

Find the equation of a straight line  with slope − 1/3 and y-intercept − 4.

Ex. 23.30 | Q 2.3 | Page 21

Find the equation of a straight line with slope −2 and intersecting the x-axis at a distance of 3 units to the left of origin.

Ex. 23.30 | Q 3 | Page 21

Find the equations of the bisectors of the angles between the coordinate axes.

Ex. 23.30 | Q 4 | Page 21

Find the equation of a line which makes an angle of tan−1 (3) with the x-axis and cuts off an intercept of 4 units on negative direction of y-axis.

Ex. 23.30 | Q 5 | Page 21

Find the equation of a line that has y-intercept −4 and is parallel to the line joining (2, −5) and (1, 2).

Ex. 23.30 | Q 6 | Page 21

Find the equation of a line which is perpendicular to the line joining (4, 2) and (3, 5) and cuts off an intercept of length 3 on y-axis.

Ex. 23.30 | Q 7 | Page 21

Find the equation of the perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-axis.

Ex. 23.30 | Q 8 | Page 21

Find the equation of the strainght line intersecting y-axis at a distance of 2 units above the origin and making an angle of 30° with the positive direction of the x-axis.

Chapter 23: The straight lines Exercise 23.40 solutions [Page 29]

Ex. 23.40 | Q 1 | Page 29

Find the equation of the straight line passing through the point (6, 2) and having slope − 3.

Ex. 23.40 | Q 2 | Page 29

Find the equation of the straight line passing through (−2, 3) and inclined at an angle of 45° with the x-axis.

Ex. 23.40 | Q 3 | Page 29

Find the equation of the line passing through (0, 0) with slope m.

Ex. 23.40 | Q 4 | Page 29

Find the equation of the line passing through $(2, 2\sqrt{3})$ and inclined with x-axis at an angle of 75°.

Ex. 23.40 | Q 5 | Page 29

Find the equation of the straight line which passes through the point (1,2) and makes such an angle with the positive direction of x-axis whose sine is $\frac{3}{5}$.

Ex. 23.40 | Q 6 | Page 29

Find the equation of the straight line passing through (3, −2) and making an angle of 60° with the positive direction of y-axis.

Ex. 23.40 | Q 7 | Page 29

Find the lines through the point (0, 2) making angles $\frac{\pi}{3} \text { and } \frac{2\pi}{3}$  with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.

Ex. 23.40 | Q 8 | Page 29

Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.

Ex. 23.40 | Q 9 | Page 29

Find the equation of the line which intercepts a length 2 on the positive direction of the x-axis and is inclined at an angle of 135° with the positive direction of y-axis.

Ex. 23.40 | Q 10 | Page 29

Find the equation of the straight line which divides the join of the points (2, 3) and (−5, 8) in the ratio 3 : 4 and is also perpendicular to it.

Ex. 23.40 | Q 11 | Page 29

Prove that the perpendicular drawn from the point (4, 1) on the join of (2, −1) and (6, 5) divides it in the ratio 5 : 8.

Ex. 23.40 | Q 12 | Page 29

Find the equations to the altitudes of the triangle whose angular points are A (2, −2), B (1, 1) and C (−1, 0).

Ex. 23.40 | Q 13 | Page 29

Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).

Ex. 23.40 | Q 14 | Page 29

Find the equation of the line passing through the point (−3, 5) and perpendicular to the line joining (2, 5) and (−3, 6).

Ex. 23.40 | Q 15 | Page 29

Find the equation of the right bisector of the line segment joining the points A (1, 0) and B (2, 3).

Chapter 23: The straight lines Exercise 23.50 solutions [Pages 35 - 36]

Ex. 23.50 | Q 1.1 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(0, 0) and (2, −2)

Ex. 23.50 | Q 1.2 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(a, b) and (a + c sin α, b + c cos α)

Ex. 23.50 | Q 1.3 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(0, −a) and (b, 0)

Ex. 23.50 | Q 1.4 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(a, b) and (a + b, a − b)

Ex. 23.50 | Q 1.5 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(at1, a/t1) and (at2, a/t2)

Ex. 23.50 | Q 1.6 | Page 35

Find the equation of the straight lines passing through the following pair of point :

(a cos α, a sin α) and (a cos β, a sin β)

Ex. 23.50 | Q 2.1 | Page 35

Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (1, 4), (2, −3) and (−1, −2).

Ex. 23.50 | Q 2.2 | Page 35

Find the equations of the sides of the triangles the coordinates of whose angular point is  respectively  (0, 1), (2, 0) and (−1, −2).

Ex. 23.50 | Q 3 | Page 35

Find the equations of the medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).

Ex. 23.50 | Q 4 | Page 35

Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a', y= b and y = b'.

Ex. 23.50 | Q 5 | Page 35

Find the equation of the side BC of the triangle ABC whose vertices are (−1, −2), (0, 1) and (2, 0) respectively. Also, find the equation of the median through (−1, −2).

Ex. 23.50 | Q 6 | Page 35

By using the concept of equation of a line, prove that the three points (−2, −2), (8, 2) and (3, 0) are collinear.

Ex. 23.50 | Q 7 | Page 35

Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.

Ex. 23.50 | Q 8 | Page 35

Find the equation to the straight line which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (−a, b) and (a', −b').

Ex. 23.50 | Q 9 | Page 35

In what ratio is the line joining the points (2, 3) and (4, −5) divided by the line passing through the points (6, 8) and (−3, −2).

Ex. 23.50 | Q 10 | Page 35

The vertices of a quadrilateral are A (−2, 6), B (1, 2), C (10, 4) and D (7, 8). Find the equation of its diagonals.

Ex. 23.50 | Q 11 | Page 35

The length L (in centimeters) of a copper rod is a linear function of its celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L in terms of C.

Ex. 23.50 | Q 12 | Page 35

The owner of a milk store finds that he can sell 980 litres milk each week at Rs 14 per liter and 1220 liters of milk each week at Rs 16 per liter. Assuming a linear relationship between selling price and demand, how many liters could he sell weekly at Rs 17 per liter.

Ex. 23.50 | Q 13 | Page 35

Find the equation of the bisector of angle A of the triangle whose vertices are A (4, 3), B (0, 0) and C(2, 3).

Ex. 23.50 | Q 14 | Page 35

Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3 x + y = 12 which is intercepted between the axes of coordinates.

Ex. 23.50 | Q 15 | Page 36

Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y =1.

Chapter 23: The straight lines Exercise 23.60 solutions [Pages 46 - 47]

Ex. 23.60 | Q 1.1 | Page 46

Find the equation to the straight line cutting off intercepts 3 and 2 from the axes.

Ex. 23.60 | Q 1.2 | Page 46

Find the equation to the straight line cutting off intercepts − 5 and 6 from the axes.

Ex. 23.60 | Q 2 | Page 46

Find the equation of the straight line which passes through (1, −2) and cuts off equal intercepts on the axes.

Ex. 23.60 | Q 3 | Page 46

Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes
(i) equal in magnitude and both positive,
(ii) equal in magnitude but opposite in sign.

Ex. 23.60 | Q 4 | Page 46

For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x − 3y + 6 = 0 on the axes.

Ex. 23.60 | Q 5 | Page 47

Find the equation to the straight line which cuts off equal positive intercepts on the axes and their product is 25.

Ex. 23.60 | Q 6 | Page 47

Find the equation of the line which passes through the point (− 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.

Ex. 23.60 | Q 7 | Page 47

A straight line passes through the point (α, β) and this point bisects the portion of the line intercepted between the axes. Show that the equation of the straight line is $\frac{x}{2 \alpha} + \frac{y}{2 \beta} = 1$.

Ex. 23.60 | Q 8 | Page 47

Find the equation of the line which passes through the point (3, 4) and is such that the portion of it intercepted between the axes is divided by the point in the ratio 2:3.

Ex. 23.60 | Q 9 | Page 47

Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find the equation of the line.

Ex. 23.60 | Q 10 | Page 47

Find the equation of the straight line which passes through the point (−3, 8) and cuts off positive intercepts on the coordinate axes whose sum is 7.

Ex. 23.60 | Q 12 | Page 47

Find the equation of a line which passes through the point (22, −6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.

Ex. 23.60 | Q 13 | Page 47

Find the equation of the line, which passes through P (1, −7) and meets the axes at A and Brespectively so that 4 AP − 3 BP = 0.

Ex. 23.60 | Q 14 | Page 47

Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.

Ex. 23.60 | Q 15 | Page 47

Find the equation of the straight line which passes through the point P (2, 6) and cuts the coordinate axes at the point A and B respectively so that $\frac{AP}{BP} = \frac{2}{3}$ .

Ex. 23.60 | Q 16 | Page 47

Find the equations of the straight lines each of which passes through the point (3, 2) and cuts off intercepts a and b respectively on X and Y-axes such that a − b = 2.

Ex. 23.60 | Q 17 | Page 47

Find the equations of the straight lines which pass through the origin and trisect the portion of the straight line 2x + 3y = 6 which is intercepted between the axes.

Ex. 23.60 | Q 18 | Page 47

Find the equation of the straight line passing through the point (2, 1) and bisecting the portion of the straight line 3x − 5y = 15 lying between the axes.

Ex. 23.60 | Q 19 | Page 47

Find the equation of the straight line passing through the origin and bisecting the portion of the line ax + by + c = 0 intercepted between the coordinate axes.

Chapter 23: The straight lines Exercise 23.70 solutions [Pages 53 - 54]

Ex. 23.70 | Q 1.1 | Page 53

Find the equation of a line for  p = 5, α = 60°.

Ex. 23.70 | Q 1.2 | Page 53

Find the equation of a line for p = 4, α = 150°.

Ex. 23.70 | Q 1.3 | Page 53

Find the equation of a line for p = 8, α = 225°.

Ex. 23.70 | Q 1.4 | Page 53

Find the equation of a line for p = 8, α = 300°.

Ex. 23.70 | Q 2 | Page 53

Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.

Ex. 23.70 | Q 3 | Page 53

Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x-axis is 15°.

Ex. 23.70 | Q 4 | Page 53

Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle tan−1 $\left( \frac{5}{12} \right)$ with the positive direction of x-axi .

Ex. 23.70 | Q 5 | Page 53

Find the equation of the straight line on which the length of the perpendicular from the origin is 2 and the perpendicular makes an angle α with x-axis such that sin α = $\frac{1}{3}$.

Ex. 23.70 | Q 6 | Page 53

Find the equation of the straight line upon which the length of the perpendicular from the origin is 2 and the slope of this perpendicular is $\frac{5}{12}$.

Ex. 23.70 | Q 7 | Page 53

The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150° with the positive direction of Y-axis. Find the equation of the line.

Ex. 23.70 | Q 8 | Page 53

Find the value of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line $\sqrt{3}x + y + 2 = 0$.

Ex. 23.70 | Q 9 | Page 53

Find the equation of the straight line which makes a triangle of area $96\sqrt{3}$ with the axes and perpendicular from the origin to it makes an angle of 30° with Y-axis.

Ex. 23.70 | Q 10 | Page 54

Find the equation of a straight line on which the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area $50/\sqrt{3}$ with the axes.

Chapter 23: The straight lines Exercise 23.80 solutions [Pages 65 - 66]

Ex. 23.80 | Q 1 | Page 65

A line passes through a point A (1, 2) and makes an angle of 60° with the x-axis and intersects the line x + y = 6 at the point P. Find AP.

Ex. 23.80 | Q 2 | Page 65

If the straight line through the point P (3, 4) makes an angle π/6 with the x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.

Ex. 23.80 | Q 3 | Page 65

A straight line drawn through the point A (2, 1) making an angle π/4 with positive x-axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.

Ex. 23.80 | Q 4 | Page 65

A line a drawn through A (4, −1) parallel to the line 3x − 4y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

Ex. 23.80 | Q 5 | Page 65

The straight line through P (x1, y1) inclined at an angle θ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.

Ex. 23.80 | Q 6 | Page 66

Find the distance of the point (2, 3) from the line 2x − 3y + 9 = 0 measured along a line making an angle of 45° with the x-axis.

Ex. 23.80 | Q 7 | Page 66

Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to a line having slope 1/2.

Ex. 23.80 | Q 8 | Page 66

Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to a line having slope 3/4.

Ex. 23.80 | Q 9 | Page 66

Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x − 2y = 1.

Ex. 23.80 | Q 10 | Page 66

Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y+ 8 = 0.

Ex. 23.80 | Q 11 | Page 66

Find the distance of the line 2x + y = 3 from the point (−1, −3) in the direction of the line whose slope is 1.

Ex. 23.80 | Q 12 | Page 66

A line is such that its segment between the straight lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0 is bisected at the point (1, 5). Obtain its equation.

Ex. 23.80 | Q 13 | Page 66

Find the equation of straight line passing through (−2, −7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3.

Chapter 23: The straight lines Exercise 23.90 solutions [Page 72]

Ex. 23.90 | Q 1.1 | Page 72

Reduce the equation $\sqrt{3}$ x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;

Ex. 23.90 | Q 1.2 | Page 72

Reduce the equation$\sqrt{3}$ x + y + 2 = 0 to intercept form and find intercept on the axes.

Ex. 23.90 | Q 1.3 | Page 72

Reduce the equation $\sqrt{3}$ x + y + 2 = 0 to the normal form and find p and α.

Ex. 23.90 | Q 2.1 | Page 72

Reduce the following equation to the normal form and find p and α in $x + \sqrt{3}y - 4 = 0$ .

Ex. 23.90 | Q 2.2 | Page 72

Reduce the following equation to the normal form and find p and α in $x + y + \sqrt{2} = 0$.

Ex. 23.90 | Q 2.3 | Page 72

Reduce the following equation to the normal form and find p and α in $x - y + 2\sqrt{2} = 0$.

Ex. 23.90 | Q 2.4 | Page 72

Reduce the following equation to the normal form and find p and α in x − 3 = 0.

Ex. 23.90 | Q 2.5 | Page 72

Reduce the following equation to the normal form and find p and α in y − 2 = 0.

Ex. 23.90 | Q 3 | Page 72

Put the equation $\frac{x}{a} + \frac{y}{b} = 1$ to the slope intercept form and find its slope and y-intercept.

Ex. 23.90 | Q 4 | Page 72

Reduce the lines 3 x − 4 y + 4 = 0 and 2 x + 4 y − 5 = 0 to the normal form and hence find which line is nearer to the origin.

Ex. 23.90 | Q 5 | Page 72

Show that the origin is equidistant from the lines 4x + 3y + 10 = 0; 5x − 12y + 26 = 0 and 7x + 24y = 50.

Ex. 23.90 | Q 6 | Page 72

Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line $\sqrt{3}x + y + 2 = 0$.

Ex. 23.90 | Q 7 | Page 72

Reduce the equation 3x − 2y + 6 = 0 to the intercept form and find the x and y intercepts.

Ex. 23.90 | Q 8 | Page 72

The perpendicular distance of a line from the origin is 5 units and its slope is − 1. Find the equation of the line.

Chapter 23: The straight lines Exercise 23.10 solutions [Pages 77 - 78]

Ex. 23.10 | Q 1.1 | Page 77

Find the point of intersection of the following pairs of lines:

2x − y + 3 = 0 and x + y − 5 = 0

Ex. 23.10 | Q 1.2 | Page 77

Find the point of intersection of the following pairs of lines:

bx + ay = ab and ax + by = ab.

Ex. 23.10 | Q 1.3 | Page 77

Find the point of intersection of the following pairs of lines:

$y = m_1 x + \frac{a}{m_1} \text { and }y = m_2 x + \frac{a}{m_2} .$

Ex. 23.10 | Q 2.1 | Page 77

Find the coordinates of the vertices of a triangle, the equations of whose sides are x + y − 4 = 0, 2x − y + 3 = 0 and x − 3y + 2 = 0.

Ex. 23.10 | Q 2.2 | Page 77

Find the coordinates of the vertices of a triangle, the equations of whose sides are

y (t1 + t2) = 2x + 2a t1t2, y (t2 + t3) = 2x + 2a t2t3 and, y (t3 + t1) = 2x + 2a t1t3.

Ex. 23.10 | Q 3.1 | Page 78

Find the area of the triangle formed by the line y = m1 x + c1, y = m2 x + c2 and x = 0.

Ex. 23.10 | Q 3.2 | Page 78

Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.

Ex. 23.10 | Q 3.3 | Page 78

Find the area of the triangle formed by the line x + y − 6 = 0, x − 3y − 2 = 0 and 5x − 3y + 2 = 0.

Ex. 23.10 | Q 4 | Page 78

Find the equations of the medians of a triangle, the equations of whose sides are:
3x + 2y + 6 = 0, 2x − 5y + 4 = 0 and x − 3y − 6 = 0

Ex. 23.10 | Q 5 | Page 78

Prove that the lines  $y = \sqrt{3}x + 1, y = 4 \text { and } y = - \sqrt{3}x + 2$ form an equilateral triangle.

Ex. 23.10 | Q 6.1 | Page 78

Classify the following pair of line as coincident, parallel or intersecting:

2x + y − 1 = 0 and 3x + 2y + 5 = 0

Ex. 23.10 | Q 6.2 | Page 78

Classify the following pair of line as coincident, parallel or intersecting:

x − y = 0 and 3x − 3y + 5 = 0]

Ex. 23.10 | Q 6.3 | Page 78

Classify the following pair of line as coincident, parallel or intersecting:

3x + 2y − 4 = 0 and 6x + 4y − 8 = 0.

Ex. 23.10 | Q 7 | Page 78

Find the equation of the line joining the point (3, 5) to the point of intersection of the lines 4x + y − 1 = 0 and 7x − 3y − 35 = 0.

Ex. 23.10 | Q 8 | Page 78

Find the equation of the line passing through the point of intersection of the lines 4x − 7y − 3 = 0 and 2x − 3y + 1 = 0 that has equal intercepts on the axes.

Ex. 23.10 | Q 9 | Page 78

Show that the area of the triangle formed by the lines y = m1 x, y = m2 x and y = c is equal to $\frac{c^2}{4}\left( \sqrt{33} + \sqrt{11} \right),$ where m1, m2 are the roots of the equation $x^2 + \left( \sqrt{3} + 2 \right)x + \sqrt{3} - 1 = 0 .$

Ex. 23.10 | Q 10 | Page 78

If the straight line $\frac{x}{a} + \frac{y}{b} = 1$ passes through the point of intersection of the lines x + y = 3 and 2x − 3y = 1 and is parallel to x − y − 6 = 0, find a and b.

Ex. 23.10 | Q 11 | Page 78

Find the orthocentre of the triangle the equations of whose sides are x + y = 1, 2x + 3y = 6 and 4x − y + 4 = 0.

Ex. 23.10 | Q 12 | Page 78

Three sides AB, BC and CA of a triangle ABC are 5x − 3y + 2 = 0, x − 3y − 2 = 0 and x + y − 6 = 0 respectively. Find the equation of the altitude through the vertex A.

Ex. 23.10 | Q 13 | Page 78

Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).

Ex. 23.10 | Q 14 | Page 78

Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x− 4y = 0, 12y + 5x = 0 and y − 15 = 0.

Ex. 23.10 | Q 15 | Page 78

Prove that the lines $\sqrt{3}x + y = 0, \sqrt{3}y + x = 0, \sqrt{3}x + y = 1 \text { and } \sqrt{3}y + x = 1$  form a rhombus.

Ex. 23.10 | Q 16 | Page 78

Find the equation of the line passing through the intersection of the lines 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.

Ex. 23.10 | Q 17 | Page 78

Find the equation of the straight line passing through the point of intersection of the lines 5x − 6y − 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x − 5y + 11 = 0 .

Chapter 23: The straight lines Exercise 23.11 solutions [Page 83]

Ex. 23.11 | Q 1.1 | Page 83

Prove that the following sets of three lines are concurrent:

15x − 18y + 1 = 0, 12x + 10y − 3 = 0 and 6x + 66y − 11 = 0

Ex. 23.11 | Q 1.2 | Page 83

Prove that the following sets of three lines are concurrent:

3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0

Ex. 23.11 | Q 1.3 | Page 83

Prove that the following sets of three lines are concurrent:

$\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1\text { and } y = x .$

Ex. 23.11 | Q 2 | Page 83

For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?

Ex. 23.11 | Q 3 | Page 83

Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.

Ex. 23.11 | Q 4 | Page 83

If the lines p1 x + q1 y = 1, p2 x + q2 y = 1 and p3 x + q3 y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

Ex. 23.11 | Q 5 | Page 83

Show that the straight lines L1 = (b + c) x + ay + 1 = 0, L2 = (c + a) x + by + 1 = 0 and L3 = (a + b) x + cy + 1 = 0 are concurrent.

Ex. 23.11 | Q 6 | Page 83

If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.

Ex. 23.11 | Q 7 | Page 83

If a, b, c are in A.P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.

Ex. 23.11 | Q 8 | Page 83

Show that the perpendicular bisectors of the sides of a triangle are concurrent.

Chapter 23: The straight lines Exercise 23.12 solutions [Pages 92 - 93]

Ex. 23.12 | Q 1 | Page 92

Find the equation of a line passing through the point (2, 3) and parallel to the line 3x − 4y + 5 = 0.

Ex. 23.12 | Q 2 | Page 92

Find the equation of a line passing through (3, −2) and perpendicular to the line x − 3y + 5 = 0.

Ex. 23.12 | Q 3 | Page 92

Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).

Ex. 23.12 | Q 4 | Page 92

Find the equations of the altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).

Ex. 23.12 | Q 5 | Page 92

Find the equation of a line which is perpendicular to the line $\sqrt{3}x - y + 5 = 0$ and which cuts off an intercept of 4 units with the negative direction of y-axis.

Ex. 23.12 | Q 6 | Page 92

If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.

Ex. 23.12 | Q 7 | Page 92

Find the equation of the straight line through the point (α, β) and perpendicular to the line lx + my + n = 0.

Ex. 23.12 | Q 8 | Page 93

Find the equation of the straight line perpendicular to 2x − 3y = 5 and cutting off an intercept 1 on the positive direction of the x-axis.

Ex. 23.12 | Q 9 | Page 93

Find the equation of the straight line perpendicular to 5x − 2y = 8 and which passes through the mid-point of the line segment joining (2, 3) and (4, 5).

Ex. 23.12 | Q 10 | Page 93

Find the equation of the straight line which has y-intercept equal to $\frac{4}{3}$ and is perpendicular to 3x − 4y + 11 = 0.

Ex. 23.12 | Q 11 | Page 93

Find the equation of the right bisector of the line segment joining the points (a, b) and (a1, b1).

Ex. 23.12 | Q 12 | Page 93

Find the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0.

Ex. 23.12 | Q 13 | Page 93

If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.

Ex. 23.12 | Q 14 | Page 93

Find the equation to the straight line parallel to 3x − 4y + 6 = 0 and passing through the middle point of the join of points (2, 3) and (4, −1).

Ex. 23.12 | Q 15 | Page 93

Prove that the lines 2x − 3y + 1 = 0, x + y = 3, 2x − 3y = 2  and x + y = 4 form a parallelogram.

Ex. 23.12 | Q 16 | Page 93

Find the equation of a line drawn perpendicular to the line $\frac{x}{4} + \frac{y}{6} = 1$ through the point where it meets the y-axis.

Ex. 23.12 | Q 17 | Page 93

The perpendicular from the origin to the line y = mx + c meets it at the point (−1, 2). Find the values of m and c.

Ex. 23.12 | Q 18 | Page 93

Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).

Ex. 23.12 | Q 19 | Page 93

The line through (h, 3) and (4, 1) intersects the line 7x − 9y − 19 = 0 at right angle. Find the value of h.

Ex. 23.12 | Q 20 | Page 93

Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.

Ex. 23.12 | Q 21 | Page 93

Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.

Ex. 23.12 | Q 22 | Page 93

Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).

Ex. 23.12 | Q 23 | Page 93

Find the equation of a line perpendicular to the line $\sqrt{3}x - y + 5 = 0$ and at a distance of 3 units from the origin.

Ex. 23.12 | Q 24 | Page 93

The line 2x + 3y = 12 meets the x-axis at A and y-axis at B. The line through (5, 5) perpendicular to AB meets the x-axis and the line AB at C and E respectively. If O is the origin of coordinates, find the area of figure OCEB.

Ex. 23.12 | Q 25 | Page 93

Find the equation of the straight line which cuts off intercepts on x-axis twice that on y-axis and is at a unit distance from the origin.

Ex. 23.12 | Q 26 | Page 93

The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively. If the point A is (1, −2), find the equation of the line BC.

Chapter 23: The straight lines Exercise 23.13 solutions [Page 99]

Ex. 23.13 | Q 1.1 | Page 99

Find the angles between the following pair of straight lines:

3x + y + 12 = 0 and x + 2y − 1 = 0

Ex. 23.13 | Q 1.2 | Page 99

Find the angles between the following pair of straight lines:

3x − y + 5 = 0 and x − 3y + 1 = 0

Ex. 23.13 | Q 1.3 | Page 99

Find the angles between the following pair of straight lines:

3x + 4y − 7 = 0 and 4x − 3y + 5 = 0

Ex. 23.13 | Q 1.4 | Page 99

Find the angles between the following pair of straight lines:

x − 4y = 3 and 6x − y = 11

Ex. 23.13 | Q 1.5 | Page 99

Find the angles between the following pair of straight lines:

(m2 − mn) y = (mn + n2) x + n3 and (mn + m2) y = (mn − n2) x + m3.

Ex. 23.13 | Q 2 | Page 99

Find the acute angle between the lines 2x − y + 3 = 0 and x + y + 2 = 0.

Ex. 23.13 | Q 3 | Page 99

Prove that the points (2, −1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.

Ex. 23.13 | Q 4 | Page 99

Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1.

Ex. 23.13 | Q 5 | Page 99

If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that  $\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}$.

Ex. 23.13 | Q 6 | Page 99

Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan−1 $\left( \frac{a}{b} \right)$.

Ex. 23.13 | Q 7 | Page 99

Find the angle between the lines x = a and by + c = 0..

Ex. 23.13 | Q 8 | Page 99

Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.

Ex. 23.13 | Q 9 | Page 99

Show that the line a2x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.

Ex. 23.13 | Q 10 | Page 99

Show that the tangent of an angle between the lines $\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}$.

Chapter 23: The straight lines Exercise 23.14 solutions [Page 102]

Ex. 23.14 | Q 1 | Page 102

Find the values of α so that the point P (α2, α) lies inside or on the triangle formed by the lines x − 5y+ 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.

Ex. 23.14 | Q 2 | Page 102

Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y − 4 = 0, 3x − 7y − 8 = 0 and 4x − y − 31 = 0.

Ex. 23.14 | Q 3 | Page 102

Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .

Chapter 23: The straight lines Exercise 23.15 solutions [Pages 107 - 108]

Ex. 23.15 | Q 1 | Page 107

Find the distance of the point (4, 5) from the straight line 3x − 5y + 7 = 0.

Ex. 23.15 | Q 2 | Page 107

Find the perpendicular distance of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin.

Ex. 23.15 | Q 3 | Page 107

Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin  β).

Ex. 23.15 | Q 4 | Page 108

Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.

Ex. 23.15 | Q 5 | Page 108

Find the distance of the point of intersection of the lines 2x + 3y = 21 and 3x − 4y + 11 = 0 from the line 8x + 6y + 5 = 0.

Ex. 23.15 | Q 6 | Page 108

Find the length of the perpendicular from the point (4, −7) to the line joining the origin and the point of intersection of the lines 2x − 3y + 14 = 0 and 5x + 4y − 7 = 0.

Ex. 23.15 | Q 7 | Page 108

What are the points on X-axis whose perpendicular distance from the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is a ?

Ex. 23.15 | Q 8 | Page 108

Show that the product of perpendiculars on the line $\frac{x}{a} \cos \theta + \frac{y}{b} \sin \theta = 1$  from the points $( \pm \sqrt{a^2 - b^2}, 0) \text { is }b^2 .$

Ex. 23.15 | Q 9 | Page 108

Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line $x - \sqrt{3}y + 4 = 0 .$

Ex. 23.15 | Q 10 | Page 108

Find the distance of the point (1, 2) from the straight line with slope 5 and passing through the point of intersection of x + 2y = 5 and x − 3y = 7.

Ex. 23.15 | Q 11 | Page 108

What are the points on y-axis whose distance from the line $\frac{x}{3} + \frac{y}{4} = 1$  is 4 units?

Ex. 23.15 | Q 12 | Page 108

In the triangle ABC with vertices A (2, 3), B (4, −1) and C (1, 2), find the equation and the length of the altitude from the vertex A.

Ex. 23.15 | Q 13 | Page 108

Show that the path of a moving point such that its distances from two lines 3x − 2y = 5 and 3x + 2y = 5 are equal is a straight line.

Ex. 23.15 | Q 14 | Page 108

If sum of perpendicular distances of a variable point P (xy) from the lines x + y − 5 = 0 and 3x − 2y + 7 = 0 is always 10. Show that P must move on a line.

Ex. 23.15 | Q 15 | Page 108

If the length of the perpendicular from the point (1, 1) to the line ax − by + c = 0 be unity, show that $\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}$ .

Chapter 23: The straight lines Exercise 23.16 solutions [Page 114]

Ex. 23.16 | Q 1.1 | Page 114

Determine the distance between the pair of parallel lines:

4x − 3y − 9 = 0 and 4x − 3y − 24 = 0

Ex. 23.16 | Q 1.2 | Page 114

Determine the distance between the pair of parallel lines:

8x + 15y − 34 = 0 and 8x + 15y + 31 = 0

Ex. 23.16 | Q 1.3 | Page 114

Determine the distance between the pair of parallel lines:

y = mx + c and y = mx + d

Ex. 23.16 | Q 1.4 | Page 114

Determine the distance between the pair of parallel lines:

4x + 3y − 11 = 0 and 8x + 6y = 15

Ex. 23.16 | Q 2 | Page 114

The equations of two sides of a square are 5x − 12y − 65 = 0 and 5x − 12y + 26 = 0. Find the area of the square.

Ex. 23.16 | Q 3 | Page 114

Find the equation of two straight lines which are parallel to + 7y + 2 = 0 and at unit distance from the point (1, −1).

Ex. 23.16 | Q 4 | Page 114

Prove that the lines 2x + 3y = 19 and 2x + 3y + 7 = 0 are equidistant from the line 2x + 3y= 6.

Ex. 23.16 | Q 5 | Page 114

Find the equation of the line mid-way between the parallel lines 9x + 6y − 7 = 0 and 3x + 2y + 6 = 0.

Ex. 23.16 | Q 6 | Page 114

Find the ratio in which the line 3x + 4+ 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0

Chapter 23: The straight lines Exercise 23.17 solutions [Page 117]

Ex. 23.17 | Q 1 | Page 117

Prove that the area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1yd1 = 0, a2x + b2y + c2 = 0, a2x + b2y + d2 = 0 is  $\left| \frac{\left( d_1 - c_1 \right)\left( d_2 - c_2 \right)}{a_1 b_2 - a_2 b_1} \right|$ sq. units.
Deduce the condition for these lines to form a rhombus.

Ex. 23.17 | Q 2 | Page 117

Prove that the area of the parallelogram formed by the lines 3x − 4y + a = 0, 3x − 4y + 3a = 0, 4x − 3y− a = 0 and 4x − 3y − 2a = 0 is $\frac{2}{7} a^2$ sq. units..

Ex. 23.17 | Q 3 | Page 117

Show that the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n' = 0, mx + ly + n = 0 and mx + ly + n' = 0 include an angle π/2.

Chapter 23: The straight lines Exercise 23.18 solutions [Pages 124 - 125]

Ex. 23.18 | Q 1 | Page 124

Find the equation of the straight lines passing through the origin and making an angle of 45° with the straight line $\sqrt{3}x + y = 11$.

Ex. 23.18 | Q 2 | Page 124

Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line $x + y + \sqrt{3}\left( y - x \right) = a$.

Ex. 23.18 | Q 3 | Page 124

Find the equations of the straight lines passing through (2, −1) and making an angle of 45° with the line 6x + 5y − 8 = 0.

Ex. 23.18 | Q 4 | Page 124

Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle tan−1 m to the straight line y = mx + c.

Ex. 23.18 | Q 5 | Page 125

Find the equations to the straight lines passing through the point (2, 3) and inclined at and angle of 45° to the line 3x + y − 5 = 0.

Ex. 23.18 | Q 6 | Page 125

Find the equations to the sides of an isosceles right angled triangle the equation of whose hypotenues is 3x + 4y = 4 and the opposite vertex is the point (2, 2).

Ex. 23.18 | Q 7 | Page 125

The equation of one side of an equilateral triangle is x − y = 0 and one vertex is $(2 + \sqrt{3}, 5)$. Prove that a second side is $y + (2 - \sqrt{3}) x = 6$  and find the equation of the third side.

Ex. 23.18 | Q 8 | Page 125

Find the equations of the two straight lines through (1, 2) forming two sides of a square of which 4x+ 7y = 12 is one diagonal.

Ex. 23.18 | Q 9 | Page 125

Find the equations of two straight lines passing through (1, 2) and making an angle of 60° with the line x + y = 0. Find also the area of the triangle formed by the three lines.

Ex. 23.18 | Q 10 | Page 125

Two sides of an isosceles triangle are given by the equations 7x − y + 3 = 0 and x + y − 3 = 0 and its third side passes through the point (1, −10). Determine the equation of the third side.

Ex. 23.18 | Q 11 | Page 125

Show that the point (3, −5) lies between the parallel lines 2x + 3y − 7 = 0 and 2x + 3y + 12 = 0 and find the equation of lines through (3, −5) cutting the above lines at an angle of 45°.

Ex. 23.18 | Q 12 | Page 125

The equation of the base of an equilateral triangle is x + y = 2 and its vertex is (2, −1). Find the length and equations of its sides.

Ex. 23.18 | Q 13 | Page 125

If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.

Chapter 23: The straight lines Exercise 23.19 solutions [Page 131]

Ex. 23.19 | Q 5 | Page 131

Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x − 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.

Ex. 23.19 | Q 6 | Page 131

Prove that the family of lines represented by x (1 + λ) + y (2 − λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.

Ex. 23.19 | Q 7 | Page 131

Show that the straight lines given by (2 + k) x + (1 + k) y = 5 + 7k for different values of k pass through a fixed point. Also, find that point.

Ex. 23.19 | Q 8 | Page 131

Find the equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 and making with the coordinate axes a triangle of area $\frac{3}{8}$ sq. units.

Ex. 23.19 | Q 9 | Page 131

Find the equation of the straight line which passes through the point of intersection of the lines 3x − y = 5 and x + 3y = 1 and makes equal and positive intercepts on the axes.

Ex. 23.19 | Q 10 | Page 131

Find the equations of the lines through the point of intersection of the lines x − 3y + 1 = 0 and 2x + 5y − 9 = 0 and whose distance from the origin is $\sqrt{5}$.

Ex. 23.19 | Q 11 | Page 131

Find the equations of the lines through the point of intersection of the lines x − y + 1 = 0 and 2x − 3y+ 5 = 0, whose distance from the point(3, 2) is 7/5.

Chapter 23: The straight lines solutions [Pages 131 - 132]

Q 1 | Page 131

Write an equation representing a pair of lines through the point (a, b) and parallel to the coordinate axes.

Q 2 | Page 132

Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.

Q 3 | Page 132

If the centroid of a triangle formed by the points (0, 0), (cos θ, sin θ) and (sin θ, − cos θ) lies on the line y = 2x, then write the value of tan θ.

Q 4 | Page 132

Write the value of θ ϵ $\left( 0, \frac{\pi}{2} \right)$ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

Q 5 | Page 132

Write the distance between the lines 4x + 3y − 11 = 0 and 8x + 6y − 15 = 0.

Q 6 | Page 132

Write the coordinates of the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.

Q 7 | Page 132

If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent, then write the value of 2abc − ab − bc − ca.

Q 8 | Page 132

Write the area of the triangle formed by the coordinate axes and the line (sec θ − tan θ) x + (sec θ + tan θ) y = 2.

Q 9 | Page 132

If the diagonals of the quadrilateral formed by the lines l1x + m1y + n1 = 0, l2x + m2y + n2 = 0, l1x + m1y + n1' = 0 and l2x + m2y + n2' = 0 are perpendicular, then write the value of l12 − l22 + m12 − m22.

Q 10 | Page 132

Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.

Q 11 | Page 132

Write the integral values of m for which the x-coordinate of the point of intersection of the lines y = mx + 1 and 3x + 4y = 9 is an integer.

Q 12 | Page 132

If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b3 − c3) x + (c3 − a3) y + (a3 − b3) = 0 represent the same line.

Q 13 | Page 132

If a, b, c are in G.P. write the area of the triangle formed by the line ax + by + c = 0 with the coordinates axes.

Q 14 | Page 132

Write the area of the figure formed by the lines a |x| + b |y| + c = 0.

Q 15 | Page 132

Write the locus of a point the sum of whose distances from the coordinates axes is unity.

Q 16 | Page 132

If a, b, c are in A.P., then the line ax + by + c = 0 passes through a fixed point. Write the coordinates of that point.

Q 17 | Page 132

Write the equation of the line passing through the point (1, −2) and cutting off equal intercepts from the axes.

Q 18 | Page 132

Find the locus of the mid-points of the portion of the line x sinθ+ y cosθ = p intercepted between the axes.

Chapter 23: The straight lines solutions [Pages 133 - 135]

Q 1 | Page 133

L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through

• (1, 1)

• (2, 1)

• (1, 2)

• none of these

Q 2 | Page 133

The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is

• $\cos^{- 1} \left( \frac{2}{3} \right)$

• $\cos^{- 1} \left( \frac{3}{4} \right)$

• $\cos^{- 1} \left( \frac{4}{5} \right)$

• $\cos^{- 1} \left( \frac{5}{6} \right)$

Q 3 | Page 133

The distance between the orthocentre and circumcentre of the triangle with vertices (1, 2), (2, 1) and $\left( \frac{3 + \sqrt{3}}{2}, \frac{3 + \sqrt{3}}{2} \right)$  is

• 0

• $\sqrt{2}$

• $3 + \sqrt{3}$

•  none of these

Q 4 | Page 133

The equation of the straight line which passes through the point (−4, 3) such that the portion of the line between the axes is divided internally by the point in the ratio 5 : 3 is

• 9x − 20y + 96 = 0

•  9x + 20y = 24

•  20x + 9y + 53 = 0

• none of these

Q 5 | Page 133

The point which divides the join of (1, 2) and (3, 4) externally in the ratio 1 : 1

•  lies in the III quadrant

• lies in the II quadrant

•  lies in the I quadrant

• cannot be found

Q 6 | Page 133

A line passes through the point (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is

• $\frac{1}{3}$

• 2/3

• 1

• 4/3

Q 7 | Page 133

If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in

•  H.P.

• G.P.

• A.P.

• none of these

Q 8 | Page 133

The number of real values of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent is

• 0

• 1

• 2

•  Infinite

Q 9 | Page 133

The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is

•  x − 3y + 1 = 0

• x − 3y + 4 = 0

• 3x − y + 2 = 0

• none of these

Q 10 | Page 133

If p1 and p2 are the lengths of the perpendiculars from the origin upon the lines x sec θ + y cosec θ = a and x cos θ − y sin θ = a cos 2 θ respectively, then

• 4p12 + p22 = a2

• p12 + 4p22 = a2

•  p12 + p22 = a2

• none of these

Q 11 | Page 133

Area of the triangle formed by the points $\left( (a + 3)(a + 4), a + 3 \right), \left( (a + 2)(a + 3), (a + 2) \right) \text { and } \left( (a + 1)(a + 2), (a + 1) \right)$

• 25a2

•  5a2

• 24a2

• none of these

Q 12 | Page 134

If a + b + c = 0, then the family of lines 3ax + by + 2c = 0 pass through fixed point

•  (2, 2/3)

• (2/3, 2)

•  (−2, 2/3)

• none of these

Q 13 | Page 134

The line segment joining the points (−3, −4) and (1, −2) is divided by y-axis in the ratio

• 1 : 3

•  2 : 3

• 3 : 1

•  3 : 2

Q 14 | Page 134

The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is

• 17

• 16

• 15

• none of these

Q 15 | Page 134

The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio

• 3 : 4

• 4 : 3

• 9 : 4

• 4 : 9

Q 16 | Page 134

If the point (5, 2) bisects the intercept of a line between the axes, then its equation is

•  5x + 2y = 20

•  2x + 5y = 20

• 5x − 2y = 20

•  2x − 5y = 20

Q 17 | Page 134

A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are four points. If ∆ DBC : ∆ ABC = 1 : 2, then x is equal to

• 11/8

•  8/11

• 3

• none of these

Q 18 | Page 134

If p be the length of the perpendicular from the origin on the line x/a + y/b = 1, then

•  p2 = a2 + b

• $p^2 = \frac{1}{a^2} + \frac{1}{b^2}$

• $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$

• none of these

Q 19 | Page 134

The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 is

•  5x + 3y − 20 = 0

• 3x − 5y + 7 = 0

• 3x − 5y + 6 = 0

• 5x + 3y + 7 = 0

Q 20 | Page 134

The figure formed by the lines ax ± by ± c = 0 is

• a rectangle

• a square

• a rhombus

•  none of these

Q 21 | Page 134

Two vertices of a triangle are (−2, −1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 square units, then the third vertex is

•  (0, 5) or, (4, 1)

• (5, 0) or, (1, 4)

•  (5, 0) or, (4, 1)

• (0, 5) or, (1, 4)

• (2, 3)

Q 22 | Page 134

The inclination of the straight line passing through the point (−3, 6) and the mid-point of the line joining the point (4, −5) and (−2, 9) is

• π/4

•  π/6

• π/3

• 3 π/4

• 5 π/6

Q 23 | Page 134

Distance between the lines 5x + 3y − 7 = 0 and 15x + 9y + 14 = 0 is

• $\frac{35}{\sqrt{34}}$

• $\frac{1}{3\sqrt{34}}$

• $\frac{35}{3\sqrt{34}}$

• $\frac{35}{2\sqrt{34}}$

•  35

Q 24 | Page 134

The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is

•  90°

•  60°

•  45°

•  30°

•  180°

Q 25 | Page 134

The value of λ for which the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point is

• 2

• 1

• 4

• 3

• 0

Q 26 | Page 134

Three vertices of a parallelogram taken in order are (−1, −6), (2, −5) and (7, 2). The fourth vertex is

•  (1, 4)

• (4, 1)

•  (1, 1)

•  (4, 4)

•  (0, 0)

Q 27 | Page 135

The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is

• (0, 0)

• (4, 7)

•  (7, 4)

• (7, 7)

• (4, 4)

Q 28 | Page 135

If the lines x + q = 0, y − 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be

• 1

• 2

• 3

• 5

Q 29 | Page 135

The medians AD and BE of a triangle with vertices A (0, b), B (0, 0) and C (a, 0) are perpendicular to each other, if

• $a = \frac{b}{2}$

• $b = \frac{a}{2}$

• ab = 1

• $a = \pm \sqrt{2}b$

Q 30 | Page 135

The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is

•  3x + 2y − 63 = 0

•  3x + 2y − 2 = 0

• 2y − 3x − 2 = 0

• none of these

Q 31 | Page 135

The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is

• $2\sqrt{2}$

• 2

• $\sqrt{2}$

• 1

Q 32 | Page 135

A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is

• (1, −1)

•  (1, 1)

• (0, 0)

• (0, 1)

Q 33 | Page 135

The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0 is

• 1: 2

• 3: 7

• 2: 3

•  2: 5

Q 34 | Page 135

The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y − 11 = 0 are

• (−6, 5)

•  (5, 6)

• (−5, 6)

• (6, 5)

Q 35 | Page 135

The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is

•  (−1, −14)

• (3, 4)

• (0, 0)

• (1, 2)

Chapter 23: The straight lines

Ex. 23.10Ex. 23.20Ex. 23.30Ex. 23.40Ex. 23.50Ex. 23.60Ex. 23.70Ex. 23.80Ex. 23.90Ex. 23.11Ex. 23.12Ex. 23.13Ex. 23.14Ex. 23.15Ex. 23.16Ex. 23.17Ex. 23.18Ex. 23.19Others

RD Sharma Mathematics Class 11 RD Sharma solutions for Class 11 Mathematics chapter 23 - The straight lines

RD Sharma solutions for Class 11 Maths chapter 23 (The straight lines) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 23 The straight lines are Slope of a Line, Various Forms of the Equation of a Line, General Equation of a Line, Distance of a Point from a Line, Brief Recall of Two Dimensional Geometry from Earlier Classes, Shifting of Origin, Equation of Family of Lines Passing Through the Point of Intersection of Two Lines.

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