Class 11 Mathematics

Chapter 9 Solutions — Straight Lines

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Solutions

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Step-by-step NCERT solutions for Straight Lines (Chapter 9, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Straight Lines textbook chapter.

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All 70 questions in Straight Lines are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 9.1

Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.
The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.
3.
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.
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
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).
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
Without using 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).
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.
A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).

Exercise 9.2

Write the equations for the x-axis and y-axis.
Passing through the point (–4, 3) with slope 1/2.
Passing through (0, 0) with slope m.
Passing through (2, 2√3) and inclined with the x-axis at an angle of 75°.
Intersecting the x-axis at a distance of 3 units to the left of origin with slope –2.
Intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30° with positive direction of the x-axis.
Passing through the points (–1, 1) and (2, –4).
The vertices of △PQR are P(2, 1), Q(–2, 3) and R(4, 5). Find equation of the median through the vertex R.
Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).
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.
Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Find equation of the line through the point (0, 2) making an angle 2π/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.
The perpendicular from the origin to a line meets it at the point (–2, 9), find the equation of the line.
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.
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?
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.
Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find equation of the line.
By using the concept of equation of a line, prove that the three points (3, 0), (–2, –2) and (8, 2) are collinear.

Exercise 9.3

1.
Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts.
- (i) x + 7y = 0,
- (ii) 6x + 3y – 5 = 0,
- (iii) y = 0.
2.
Reduce the following equations into intercept form and find their intercepts on the axes.
- (i) 3x + 2y – 12 = 0,
- (ii) 4x – 3y = 6,
- (iii) 3y + 2 = 0.
Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).
Find the points on the x-axis, whose distances from the line x/3 + y/4 = 1 are 4 units.
5.
Find the distance between parallel lines
- (i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0
- (ii) l(x + y) + p = 0 and l(x + y) – r = 0.
Find equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x-intercept 3.
Find angles between the lines √3 x + y = 1 and x + √3 y = 1.
The line through the points (h, 3) and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle. Find the value of h.
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A(x – x1) + B(y – y1) = 0.
Two lines passing through the point (2, 3) intersect each other at an angle of 60°. If slope of one line is 2, find equation of the other line.
Find the equation of the right bisector of the line segment joining the points (3, 4) and (–1, 2).
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
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.
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and x sec θ + y cosec θ = k, respectively, prove that p² + 4q² = k².
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.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that 1/p² = 1/a² + 1/b².

Miscellaneous Exercise

1.
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.
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
What are the points on the y-axis whose distance from the line x/3 + y/4 = 1 is 4 units.
Find perpendicular distance from the origin to the line joining the points (cos θ, sin θ) and (cos φ, sin φ).
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.
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.
Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0.
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.
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.
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x – 2y = 3.
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 equation of the line passing through the origin and making an angle θ with the line y = mx + c is y/x = (m ± tan θ)/(1 ∓ m tan θ).
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
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.
The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (–4, 1). Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.
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.
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
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.
Prove that the product of the lengths of the perpendiculars drawn from the points (√(a²–b²), 0) and (–√(a²–b²), 0) to the line (x/a) cos θ + (y/b) sin θ = 1 is b².
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.

Sample

Worked examples

3 questions from Straight Lines, solved step by step.

Exercise 9.1

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

The base lies along the y-axis with midpoint at the origin.

So the two endpoints of the base are at (0, a) and (0, –a).

The third vertex lies on the x-axis (by symmetry about the x-axis).

Let the third vertex be (x, 0).

Since all sides are equal to 2a, the distance from (x, 0) to (0, a) must equal 2a:

√(x² + a²) = 2a

x² + a² = 4a²

x² = 3a²

x = ±a√3

So the third vertex is (a√3, 0) or (–a√3, 0).

Answer: The three vertices are (0, a), (0, –a), and (a√3, 0). If the triangle is on the other side, the third vertex is (–a√3, 0).

Exercise 9.2

Write the equations for the x-axis and y-axis.

Every point on the x-axis has y-coordinate = 0, regardless of x.

So the equation of the x-axis is: y = 0.

Every point on the y-axis has x-coordinate = 0, regardless of y.

So the equation of the y-axis is: x = 0.

Answer: Equation of the x-axis: y = 0. Equation of the y-axis: x = 0.

Exercise 9.3

Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts. (i) x + 7y = 0, (ii) 6x + 3y – 5 = 0, (iii) y = 0.

(i) x + 7y = 0

7y = –x

y = (–1/7)x + 0.

Slope m = –1/7, y-intercept c = 0.

(ii) 6x + 3y – 5 = 0

3y = –6x + 5

y = –2x + 5/3.

Slope m = –2, y-intercept c = 5/3.

(iii) y = 0

This is already in the form y = mx + c with m = 0 and c = 0.

Slope m = 0, y-intercept c = 0.

Answer: (i) Slope = –1/7, y-intercept = 0. (ii) Slope = –2, y-intercept = 5/3. (iii) Slope = 0, y-intercept = 0.

All 70 questions in Straight Lines are solved step by step in the Solutions PDF above.

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Chapter 9 Solutions — Straight Lines

Overview

What these solutions cover

Exercise 9.1

Exercise 9.2

Exercise 9.3

Miscellaneous Exercise

Worked examples

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

Write the equations for the x-axis and y-axis.

Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts. (i) x + 7y = 0, (ii) 6x + 3y – 5 = 0, (iii) y = 0.

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