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NCERT solutions Mathematics Textbook for Class 11 chapter 10 Straight Lines

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Chapter 10 - Straight Lines

Pages 211 - 212

Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.

Q 1 | Page 211 | view solution

The base of an equilateral triangle with side 2a lies along they y-axis such that the mid point of the base is at the origin. Find vertices of the triangle.

Q 2 | Page 211 | view solution

Find the distance between P (x1, y1) and Q (x2, y2) when : (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis

Q 3 | Page 211 | view solution

Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).

Q 4 | Page 211 | view solution

Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.

Q 6 | Page 212 | view solution

Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.

Q 7 | Page 212 | view solution

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

Q 8 | Page 212 | view solution

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

Q 9 | Page 212 | view solution

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

Q 10 | Page 212 | view solution

The slope of a line is double of the slope of another line. If tangent of the angle between them is `1/3`, find the slopes of the lines

Q 11 | Page 212 | view solution

A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).

Q 12 | Page 212 | view solution

If three point (h, 0), (a, b) and (0, k) lie on a line, show that `q/h + b/k = 1`

Q 13 | Page 212 | view solution

Consider the given population and year graph. Find the slope of the line AB and using it, find what will be the population in the year 2010?

Q 14 | Page 212 | view solution

Pages 219 - 220

Write the equations for the x and y-axes.

Q 1 | Page 219 | view solution

Find the equation of the line which passes through the point (–4, 3) with slope `1/2`

Q 2 | Page 219 | view solution

Find the equation of the line which passes though (0, 0) with slope m.

Q 3 | Page 219 | view solution

Find the equation of the line which passes though `(2, 2sqrt3)` and is inclined with the x-axis at an angle of 75°

Q 4 | Page 219 | view solution

Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope –2.

Q 5 | Page 219 | view solution

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

Q 6 | Page 219 | view solution

Find the equation of the line which passes through the points (–1, 1) and (2, –4).

Q 7 | Page 219 | view solution

Find the equation of the line which is at a perpendicular distance of 5 units from the origin and the angle made by the perpendicular with the positive x-axis is 30°

Q 8 | Page 220 | view solution

The vertices of ΔPQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.

Q 9 | Page 220 | view solution

The vertices of ΔPQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.

Q 9 | Page 220 | view solution

Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).

Q 10 | Page 220 | view solution

A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1:n. Find the equation of the line.

Q 11 | Page 220 | view solution

Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

Q 12 | Page 220 | view solution

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

Q 13 | Page 220 | view solution

Find equation of the line through the point (0, 2) making an angle  `(2pi)/3` with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.

Q 14 | Page 220 | view solution

The perpendicular from the origin to a line meets it at the point (– 2, 9), find the equation of the line.

Q 15 | Page 220 | view solution

The length L (in centimetre) 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

Q 16 | Page 220 | view solution

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

Q 17 | Page 220 | view solution

P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is `x/a + y/b = 2`

 
Q 18 | Page 220 | view solution

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

Q 19 | Page 220 | view solution

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

Q 20 | Page 220 | view solution

Pages 227 - 228

Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts.

(i) + 7= 0

(ii) 6+ 3– 5 = 0

(iii) = 0

 
Q 1 | Page 227 | view solution

Reduce the following equations into intercept form and find their intercepts on the axes.

(i) 3+ 2– 12 = 0

(ii) 4– 3= 6

(iii) 3+ 2 = 0

Q 2 | Page 227 | view solution

Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.

i) `x – sqrt3y + 8 = 0`

(ii) – 2 = 0

(iii) – = 4

 

Q 3 | Page 227 | view solution

Find the distance of the point (–1, 1) from the line 12(+ 6) = 5(– 2).

Q 4 | Page 227 | view solution

Find the points on the x-axis, whose distances from the `x/3 +y/4 = 1`  are 4 units.

 
Q 5 | Page 227 | view solution

Find the distance between parallel lines 15+ 8– 34 = 0 and 15+ 8+ 31 = 0

 

Q 6.1 | Page 227 | view solution

Find the distance between parallel lines  (y) + = 0 and (y) – = 0

Q 6.2 | Page 227 | view solution

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

Q 7 | Page 228 | view solution

Find equation of the line perpendicular to the line – 7+ 5 = 0 and having intercept 3.

Q 8 | Page 228 | view solution

Find angles between the lines `sqrt3x + y = 1 and  x +     sqrt3y = 1`

Q 9 | Page 228 | view solution

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

Q 10 | Page 228 | view solution

Prove that the line through the point (x1y1) and parallel to the line Ax + By + C = 0 is A (x –x1B (y – y1) = 0.

Q 11 | Page 228 | view solution

Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.

Q 12 | Page 228 | view solution

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

Q 13 | Page 228 | view solution

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

Q 14 | Page 228 | view solution

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

Q 15 | Page 228 | view solution

If and are the lengths of perpendiculars from the origin to the lines cos θ – sin θ = cos 2θ and xsec θcosec θ = k, respectively, prove that p2 + 4q2 = k2

Q 16 | Page 228 | view solution

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

Q 17 | Page 228 | view solution

If is the length of perpendicular from the origin to the line whose intercepts on the axes are and b, then show that `1/p^2 = 1/a^2 + 1/b^2`

Q 18 | Page 228 | view solution

Pages 233 - 234

Find the values of k for which the line (k–3) x – (4 – k2) y + k2 –7k + 6 = 0 is 

(a) Parallel to the x-axis,

(b) Parallel to the y-axis,

(c) Passing through the origin.

Q 1 | Page 233 | view solution

Find the values of q and p, if the equation x cos q + y sinq = p is the normal form of the line `sqrt3 x` + y + 2 = 0.

Q 2 | Page 233 | view solution

Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.

Q 3 | Page 233 | view solution

What are the points on the y-axis whose distance from the line  `x/3 + y/4 = 1` is 4 units

Q 4 | Page 233 | view solution

Find perpendicular distance from the origin to the line joining the points (cosΘ, sin Θ) and (cosΦ, sin Φ).

Q 5 | Page 233 | view solution

Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x– 7y + 5 = 0 and 3x + y = 0.

Q 6 | Page 233 | view solution

Find the equation of a line drawn perpendicular to the line `x/4 + y/6 = 1`through the point, where it meets the y-axis.

Q 7 | Page 233 | view solution

Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0.

Q 8 | Page 233 | view solution

Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.

Q 9 | Page 233 | view solution

If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.

Q 10 | Page 233 | view solution

Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.

Q 11 | Page 233 | view solution

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.

Q 12 | Page 233 | view solution

Show that the equation of the line passing through the origin and making an angle θ with the line `y = mx + c " is " y/c = (m+- tan theta)/(1 +- m tan theta)`

Q 13 | Page 234 | view solution

In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?

Q 14 | Page 234 | view solution

Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.

Q 15 | Page 234 | view solution

Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.

Q 16 | Page 234 | view solution

The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle.

Q 17 | Page 234 | view solution

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

Q 18 | Page 234 | view solution

If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

Q 19 | Page 234 | view solution

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

Q 20 | Page 234 | view solution

Find equation of the line which is equidistant from parallel lines 9+ 6y – 7 = 0 and 3x + 2y + 6 = 0.

Q 21 | Page 234 | view solution

A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

Q 22 | Page 234 | view solution

Prove that the product of the lengths of the perpendiculars drawn from the points

`(sqrt(a^2 - b^2),0)` and `(-sqrta^2-b^2,0)` to the line `x/a cos theta + y/b sin theta = 1` is `b^2`

Q 23 | Page 234 | view solution

A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y+ 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

Q 24 | Page 234 | view solution

Page 211

Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, –4) and B (8, 0).

Q 5 | Page 211 | view solution

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