#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 23: The straight lines

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

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

\[- \frac{\pi}{4}\]

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

\[\frac{2\pi}{3}\]

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

\[\frac{3\pi}{4}\]

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

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

(−3, 2) and (1, 4)

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

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

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

(3, −5), and (1, 2)

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)

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)

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)

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).

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.

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

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

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)?

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

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

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

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

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

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.

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

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

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.

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.

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

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

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.

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

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

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.

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 solutions [Page 17]

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

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

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

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

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

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

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

#### Chapter 23: The straight lines solutions [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.

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

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

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.

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

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.

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

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.

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.

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 solutions [Page 29]

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

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

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

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

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 \[\frac{3}{5}\].

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

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.

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

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.

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.

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.

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

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

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

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 solutions [Pages 35 - 36]

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

(0, 0) and (2, −2)

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)

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

(0, −a) and (b, 0)

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

(a, b) and (a + b, a − b)

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

(at_{1}, a/t_{1}) and (at_{2}, a/t_{2})

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)

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

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

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

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

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).

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

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

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').

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).

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.

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.

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.

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).

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.

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 solutions [Pages 46 - 47]

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

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

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

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.

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.

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

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.

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 \[\frac{x}{2 \alpha} + \frac{y}{2 \beta} = 1\].

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.

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

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.

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.

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.

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

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 \[\frac{AP}{BP} = \frac{2}{3}\] .

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.

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.

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.

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 solutions [Pages 53 - 54]

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

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

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

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

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°.

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°.

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 .

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 = \[\frac{1}{3}\].

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 \[\frac{5}{12}\].

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.

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

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.

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 solutions [Pages 65 - 66]

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.

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.

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.

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.

The straight line through P (x_{1}, y_{1}) inclined at an angle θ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.

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.

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

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

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

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

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

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.

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 solutions [Page 72]

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

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

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

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

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

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

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

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

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

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.

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

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

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

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 solutions [Pages 77 - 78]

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

2x − y + 3 = 0 and x + y − 5 = 0

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

bx + ay = ab and ax + by = ab.

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

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

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.

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

y (t_{1} + t_{2}) = 2x + 2a t_{1}t_{2}, y (t_{2} + t_{3}) = 2x + 2a t_{2}t_{3} and, y (t_{3} + t_{1}) = 2x + 2a t_{1}t_{3}.

Find the area of the triangle formed by the line y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and x = 0.

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

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

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

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

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

2x + y − 1 = 0 and 3x + 2y + 5 = 0

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

x − y = 0 and 3x − 3y + 5 = 0]

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

3x + 2y − 4 = 0 and 6x + 4y − 8 = 0.

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.

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.

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

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.

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

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.

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

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

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.

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.

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 solutions [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

Prove that the following sets of three lines are concurrent:

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

Prove that the following sets of three lines are concurrent:

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

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

Find the conditions that the straight lines y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and y = m_{3} x + c_{3} may meet in a point.

If the lines p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1 and p_{3} x + q_{3} y = 1 be concurrent, show that the points (p_{1}, q_{1}), (p_{2}, q_{2}) and (p_{3}, q_{3}) are collinear.

Show that the straight lines L_{1} = (b + c) x + ay + 1 = 0, L_{2} = (c + a) x + by + 1 = 0 and L_{3} = (a + b) x + cy + 1 = 0 are concurrent.

If the three lines ax + a^{2}y + 1 = 0, bx + b^{2}y + 1 = 0 and cx + c^{2}y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.

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.

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

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

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

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

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

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

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.

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

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

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.

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).

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

Find the equation of the right bisector of the line segment joining the points (a, b) and (a_{1}, b_{1}).

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

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

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).

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

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.

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.

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

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

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

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

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

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.

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.

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.

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 solutions [Page 99]

Find the angles between the following pair of straight lines:

3x + y + 12 = 0 and x + 2y − 1 = 0

Find the angles between the following pair of straight lines:

3x − y + 5 = 0 and x − 3y + 1 = 0

Find the angles between the following pair of straight lines:

3x + 4y − 7 = 0 and 4x − 3y + 5 = 0

Find the angles between the following pair of straight lines:

x − 4y = 3 and 6x − y = 11

Find the angles between the following pair of straight lines:

(m^{2} − mn) y = (mn + n^{2}) x + n^{3} and (mn + m^{2}) y = (mn − n^{2}) x + m^{3}.

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

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.

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

If θ is the angle which the straight line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].

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 }\[\left( \frac{a}{b} \right)\].

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

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

Show that the line a^{2}x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.

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 solutions [Page 102]

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

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.

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 solutions [Pages 107 - 108]

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

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

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).

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.

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.

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.

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

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

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

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.

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

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*.

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

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

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 solutions [Page 114]

Determine the distance between the pair of parallel lines:

4*x* − 3*y* − 9 = 0 and 4*x* − 3*y* − 24 = 0

Determine the distance between the pair of parallel lines:

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

Determine the distance between the pair of parallel lines:

y = mx + c and y = mx + d

Determine the distance between the pair of parallel lines:

4*x* + 3*y* − 11 = 0 and 8*x* + 6*y* = 15

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

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

#### Answer 3:

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

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

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

#### Chapter 23: The straight lines solutions [Page 117]

Prove that the area of the parallelogram formed by the lines *a*_{1}*x* + *b*_{1}*y* + *c*_{1} = 0, *a*_{1}*x* + *b*_{1}*y*+ *d*_{1} = 0, *a*_{2}*x* + *b*_{2}*y* + *c*_{2} = 0, *a*_{2}*x* + *b*_{2}*y* + *d*_{2} = 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.

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..

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 solutions [Pages 124 - 125]

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

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

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

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.

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.

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).

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.

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.

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.

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.

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°.

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.

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 solutions [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.

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.

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.

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.

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.

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 \[\sqrt{5}\].

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]

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

Write the coordinates of the orthocentre of the triangle formed by the lines x^{2} − y^{2} = 0 and x + 6y = 18.

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.

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.

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

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

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.

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.

If the diagonals of the quadrilateral formed by the lines l_{1}x + m_{1}y + n_{1} = 0, l_{2}x + m_{2}y + n_{2} = 0, l_{1}x + m_{1}y + n_{1}' = 0 and l_{2}x + m_{2}y + n_{2}' = 0 are perpendicular, then write the value of l_{1}^{2} − l_{2}^{2} + m_{1}^{2} − m_{2}^{2}.

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

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.

If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b^{3} − c^{3}) x + (c^{3} − a^{3}) y + (a^{3} − b^{3}) = 0 represent the same line.

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.

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

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

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.

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

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]

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

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

\[\cos^{- 1} \left( \frac{2}{3} \right)\]

\[\cos^{- 1} \left( \frac{3}{4} \right)\]

\[\cos^{- 1} \left( \frac{4}{5} \right)\]

\[\cos^{- 1} \left( \frac{5}{6} \right)\]

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

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

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

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

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

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

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

If p_{1} and p_{2} 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

4p

_{1}^{2}+ p_{2}^{2}= a^{2}p

_{1}^{2}^{ }+ 4p_{2}^{2}= a^{2}p

_{1}^{2}+ p_{2}^{2}= a^{2}none of these

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

25a

^{2}5a

^{2}24a

^{2}none of these

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

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

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

17

16

15

none of these

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

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

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

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

p

^{2}= a^{2}+ b^{2 }\[p^2 = \frac{1}{a^2} + \frac{1}{b^2}\]

\[\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}\]

none of these

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

5

*x*+ 3*y*− 20 = 03

*x*− 5*y*+ 7 = 03

*x*− 5*y*+ 6 = 05

*x*+ 3*y*+ 7 = 0

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

a rectangle

a square

a rhombus

none of these

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)

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

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

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

90°

60°

45°

30°

180°

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

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)

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)

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

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

\[a = \pm \sqrt{2}b\]

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

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

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)

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

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)

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

#### RD Sharma Mathematics Class 11

#### Textbook solutions for 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 Equation of Family of Lines Passing Through the Point of Intersection of Two Lines, Shifting of Origin, Brief Recall of Two Dimensional Geometry from Earlier Classes, Distance of a Point from a Line, General Equation of a Line, Various Forms of the Equation of a Line, Slope of a Line.

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