NCERT Exemplar Solutions
Class 11 - MathematicsChapter 10: STRAIGHT LINES
Class 11 - Mathematics
Short Answer Type
Question. 1
Find the equation of the straight line which passes through the point \((1, -2)\) and cuts off equal intercepts from axes.
Answer:
\(x + y + 1 = 0\)
Question. 2
Find the equation of the line passing through the point \((5, 2)\) and perpendicular to the line joining the points \((2, 3)\) and \((3, -1)\).
Answer:
\(x - 4y + 3 = 0\)
Question. 3
Find the angle between the lines \(y = (2 - \sqrt{3})(x + 5)\) and \(y = (2 + \sqrt{3})(x - 7)\).
Answer:
60° or 120°
Question. 4
Find the equation of the lines which pass through the point \((3, 4)\) and cut off intercepts from the coordinate axes such that their sum is 14.
Answer:
\(x + y = 7\)
or
\(\dfrac{x}{6} + \dfrac{y}{8} = 1\)
Question. 5
Find the points on the line \(x + y = 4\) which lie at a unit distance from the line \(4x + 3y = 10\).
Answer:
\((3, 1), (-7, 11)\)
Question. 6
Show that the tangent of an angle between the lines \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) and \(\dfrac{x}{a} - \dfrac{y}{b} = 1\) is \(\dfrac{2ab}{a^2 - b^2}\).
Answer:
\(\dfrac{2ab}{a^2 - b^2}\)
Question. 7
Find the equation of lines passing through \((1, 2)\) and making angle 30° with the y-axis.
Answer:
\(y - \sqrt{3}x - 2 + \sqrt{3} = 0\)
Question. 8
Find the equation of the line passing through the point of intersection of \(2x + y = 5\) and \(x + 3y + 8 = 0\) and parallel to the line \(3x + 4y = 7\).
Answer:
\(3x + 4y + 3 = 0\)
Question. 9
For what values of \(a\) and \(b\) are the intercepts cut off on the coordinate axes by the line \(ax + by + 8 = 0\) equal in length but opposite in signs to those cut off by the line \(2x - 3y + 6 = 0\)?
Answer:
\(a = -\dfrac{8}{3},\ b = 4\)
Question. 10
If the intercept of a line between the coordinate axes is divided by the point \((-5, 4)\) in the ratio 1 : 2, then find the equation of the line.
Answer:
\(8x - 5y + 60 = 0\)
Question. 11
Find the equation of a straight line on which the length of the perpendicular from the origin is four units and the line makes an angle of 120° with the positive direction of the x-axis.
Answer:
\(\sqrt{3}x + y = 8\)
Question. 12
Find the equation of one of the sides of an isosceles right-angled triangle whose hypotenuse is given by \(3x + 4y = 4\) and the opposite vertex of the hypotenuse is \((2, 2)\).
Answer:
\(x - 7y - 12 = 0\)
Long Answer Questions
Question. 13
If the equation of the base of an equilateral triangle is \(x + y = 2\) and the vertex is \((2, -1)\), find the length of the side of the triangle.
Answer:
The distance from the vertex \((2, -1)\) to the line \(x + y = 2\) is the altitude of the equilateral triangle.
Using the perpendicular distance formula:
\[ d = \dfrac{|2 + (-1) - 2|}{\sqrt{1^2 + 1^2}} = \dfrac{1}{\sqrt{2}} \]
In an equilateral triangle, altitude \(h = \dfrac{\sqrt{3}}{2} a\), where \(a\) is the side.
So, \( \dfrac{\sqrt{3}}{2} a = \dfrac{1}{\sqrt{2}} \Rightarrow a = \sqrt{\dfrac{2}{3}} \).
Therefore, the side length is \( \sqrt{\dfrac{2}{3}} \).
Question. 14
A variable line passes through a fixed point \(P\). The algebraic sum of the perpendiculars drawn from the points \((2,0)\), \((0,2)\), and \((1,1)\) on the line is zero. Find the coordinates of point \(P\).
Answer:
Let the line through \(P(x_1,y_1)\) be \(y - y_1 = m(x - x_1)\).
Convert to normal form and compute perpendicular distances from \((2,0)\), \((0,2)\), and \((1,1)\).
Using the condition that the algebraic sum of perpendiculars is zero gives:
\[ (2 - x_1)m - (y_1) + (0 - x_1)m + (2 - y_1) + (1 - x_1)m + (1 - y_1) = 0 \]
Simplifying yields \(x_1 = 1, y_1 = 1\).
Thus, the fixed point is \((1,1)\).
Question. 15
In what direction should a line be drawn through the point \((1, 2)\) so that its point of intersection with the line \(x + y = 4\) is at a distance \(\dfrac{\sqrt{6}}{3}\) from the given point?
Answer:
Let the required line through \((1,2)\) make angle \(\theta\) with the positive x-axis.
Its parametric form is:
\[ (x, y) = (1, 2) + t(\cos\theta, \sin\theta). \]
Substitute into \(x + y = 4\):
\[ 1 + 2 + t(\cos\theta + \sin\theta) = 4 \Rightarrow t = \dfrac{1}{\cos\theta + \sin\theta}. \]
The distance from \((1,2)\) to the intersection point is:
\[ |t| = \dfrac{1}{\cos\theta + \sin\theta} = \dfrac{\sqrt{6}}{3}. \]
Solving this gives \(\theta = 15^\circ\) or \(75^\circ\).
Question. 16
A straight line moves so that the sum of the reciprocals of its intercepts on the coordinate axes is constant. Show that the line passes through a fixed point.
Answer:
The intercept form of the line is:
\[ \dfrac{x}{a} + \dfrac{y}{b} = 1. \]
Given that
\[ \dfrac{1}{a} + \dfrac{1}{b} = k, \]
a constant.
Rewrite the line equation:
\[ bx + ay = ab. \]
Using \(ab(\dfrac{1}{a} + \dfrac{1}{b}) = abk\):
\[ b + a = abk. \]
Hence the equation becomes:
\[ bx + ay = a + b. \]
Dividing both sides by \(k\):
\[ b(x - k) + a(y - k) = 0. \]
This holds for every valid choice of \(a\) and \(b\), implying all such lines pass through the fixed point:
\[ (k, k). \]
Question. 17
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.
Answer:
Let the line intersect the axes at \((a,0)\) and \((0,b)\).
The point \((-4,3)\) divides the intercepts in the ratio \(5:3\):
\[ \dfrac{-4}{a} = \dfrac{5}{5+3} = \dfrac{5}{8}, \quad \dfrac{3}{b} = \dfrac{3}{8}. \]
Thus, \(a = -\dfrac{32}{5}\) and \(b = 8\).
The intercept form of the line becomes:
\[ \dfrac{x}{-32/5} + \dfrac{y}{8} = 1. \]
Simplifying yields the equation:
\[ 9x - 20y + 96 = 0. \]
Question. 18
Find the equations of the lines through the point of intersection of the lines \(x - y + 1 = 0\) and \(2x - 3y + 5 = 0\) and whose distance from the point \((3,2)\) is \(\dfrac{7}{5}\).
Answer:
Let the required line be:
\[ (x - y + 1) + \lambda (2x - 3y + 5) = 0. \]
Simplify to obtain:
\[ (1 + 2\lambda)x + (-1 - 3\lambda)y + (1 + 5\lambda) = 0. \]
Apply perpendicular distance formula to point \((3,2)\) and equate to \(\dfrac{7}{5}\).
Solving for \(\lambda\) produces two values, giving two lines:
\[ 3x - 4y + 6 = 0, \]
\[ 4x - 3y + 1 = 0. \]
Question. 19
If the sum of the distances of a moving point in a plane from the axes is 1, find the locus of the point.
Answer:
The distance of \((x,y)\) from the x-axis is \(|y|\), and from the y-axis is \(|x|\).
Given:
\[ |x| + |y| = 1. \]
This is the equation of a square of side 1, oriented at 45° with respect to the coordinate axes.
Question. 20
\(P_1, P_2\) are points on either of the two lines \(y - \sqrt{3}|x| = 2\) at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from \(P_1, P_2\) on the bisector of the angle between the given lines.
Answer:
The lines are:
\[ y = \sqrt{3}x + 2 \quad (x \ge 0), \]
\[ y = -\sqrt{3}x + 2 \quad (x < 0). \]
Their intersection is at \((0,2)\).
Points \(P_1, P_2\) lie 5 units away from \((0,2)\) on these lines.
The angle bisector is the y-axis. The foot of perpendicular from \((x_1,y_1)\) on the y-axis has x-coordinate 0 and y-coordinate equal to the point’s y-value.
The y-coordinates are:
\[ y = 2 \pm 5\cos 30^\circ = 2 \pm \dfrac{5\sqrt{3}}{2}. \]
Thus the feet of perpendiculars are:
\[ \left(0, 2 + \dfrac{5\sqrt{3}}{2}\right), \, \left(0, 2 - \dfrac{5\sqrt{3}}{2}\right). \]
Question. 21
If \(p\) is the length of perpendicular from the origin on the line \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) and \(a^2, p^2, b^2\) are in A.P., show that \(a^4 + b^4 = 0\).
Answer:
The perpendicular distance from origin to the line \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) is:
\[ p = \dfrac{ab}{\sqrt{a^2 + b^2}}. \]
Given \(a^2, p^2, b^2\) in A.P., so:
\[ 2p^2 = a^2 + b^2. \]
Substitute the expression for \(p^2\):
\[ 2 \left( \dfrac{a^2 b^2}{a^2 + b^2} \right) = a^2 + b^2. \]
Cross-multiplying gives:
\[ 2a^2 b^2 = (a^2 + b^2)^2. \]
Expanding the right-hand side:
\[ 2a^2 b^2 = a^4 + 2a^2 b^2 + b^4. \]
Simplifying:
\[ a^4 + b^4 = 0. \]
Objective Type Question
Question. 22
A line cutting off intercept \(-3\) from the y-axis and the tangent of angle to the x-axis is \(\tfrac{3}{?}\), its equation is
5y - 3x + 15 = 0
3y - 5x + 15 = 0
5y - 3x - 15 = 0
None of these
Question. 23
Slope of a line which cuts off intercepts of equal lengths on the axes is
-1
0
2
\(\sqrt{3}\)
Question. 24
The equation of the straight line passing through the point \((3,2)\) and perpendicular to the line \(y = x\) is
x - y = 5
x + y = 5
x + y = 1
x - y = 1
Question. 25
The equation of the line passing through the point \((1,2)\) and perpendicular to the line \(x + y + 1 = 0\) is
y - x + 1 = 0
y - x - 1 = 0
y - x + 2 = 0
y - x - 2 = 0
Question. 26
The tangent of angle between the lines whose intercepts on the axes are \(a, -b\) and \(b, -a\), respectively, is
\(\dfrac{a^2 - b^2}{ab}\)
\(\dfrac{b^2 - a^2}{2}\)
\(\dfrac{b^2 - a^2}{2ab}\)
None of these
Question. 27
If the line \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) passes through the points \((2,-3)\) and \((4,-5)\), then \((a,b)\) is
(1, 1)
(-1, 1)
(1, -1)
(-1, -1)
Question. 28
The distance of the point of intersection of the lines \(2x - 3y + 5 = 0\) and \(3x + 4y = 0\) from the line \(5x - 2y = 0\) is
\(\dfrac{130}{17\sqrt{29}}\)
\(\dfrac{13}{7\sqrt{29}}\)
\(\dfrac{130}{7}\)
None of these
Question. 29
The equations of the lines which pass through the point \((3,-2)\) and are inclined at \(60^\circ\) to the line \(\sqrt{3}\,x + y = 1\) is
\(y + 2 = 0\) and \(\sqrt{3}x - y - 2 - 3\sqrt{3} = 0\)
\(x - 2 = 0\) and \(\sqrt{3}x - y + 2 + 3\sqrt{3} = 0\)
\(\sqrt{3}x - y - 2 - 3\sqrt{3} = 0\)
None of these
Question. 30
The equations of the lines passing through the point \((1,0)\) and at a distance \(\dfrac{\sqrt{3}}{2}\) from the origin are
\(\sqrt{3}x + y - \sqrt{3} = 0\), \(\sqrt{3}x - y - \sqrt{3} = 0\)
\(\sqrt{3}x + y + \sqrt{3} = 0\), \(\sqrt{3}x - y + \sqrt{3} = 0\)
x + \sqrt{3}y - \sqrt{3} = 0, \; x - \sqrt{3}y - \sqrt{3} = 0
None of these
Question. 31
The distance between the lines \(y = mx + c_1\) and \(y = mx + c_2\) is
\(\dfrac{c_1 - c_2}{\sqrt{m^2 + 1}}\)
\(\dfrac{|c_1 - c_2|}{\sqrt{1 + m^2}}\)
\(\dfrac{c_2 - c_1}{\sqrt{1 + m^2}}\)
0
Question. 32
The coordinates of the foot of perpendiculars from the point \((2,3)\) on the line \(y = 3x + 4\) is given by
\(\left(\dfrac{37}{10}, -\dfrac{1}{10}\right)\)
\(\left(-\dfrac{1}{10}, \dfrac{37}{10}\right)\)
\(\left(\dfrac{10}{37}, -10\right)\)
\(\dfrac{2}{3}, -\dfrac{1}{3}\)
Question. 33
If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is \((3,2)\), then the equation of the line will be
2x + 3y = 12
3x + 2y = 12
4x - 3y = 6
5x - 2y = 10
Question. 34
Equation of the line passing through \((1,2)\) and parallel to the line \(y = 3x - 1\) is
y + 2 = x + 1
y + 2 = 3(x + 1)
y - 2 = 3(x - 1)
y - 2 = x - 1
Question. 35
Equations of diagonals of the square formed by the lines \(x = 0\), \(y = 0\), \(x = 1\) and \(y = 1\) are
y = x, \; y + x = 1
y = x, \; x + y = 2
2y = x, \; y + x = \dfrac{1}{3}
y = 2x, \; y + 2x = 1
Question. 36
For specifying a straight line, how many geometrical parameters should be known?
1
2
4
3
Question. 37
The point \((4,1)\) undergoes the following two successive transformations:
(i) Reflection about the line \(y = x\)
(ii) Translation through a distance 2 units along the positive x-axis
Then the final coordinates of the point are
(4, 3)
(3, 4)
(1, 4)
\(\left(\tfrac{7}{2},\tfrac{7}{2}\right)\)
Question. 38
A point equidistant from the lines \(4x + 3y + 10 = 0\), \(5x - 12y + 26 = 0\) and \(7x + 24y - 50 = 0\) is
(1, -1)
(1, 1)
(0, 0)
(0, 1)
Question. 39
A line passes through \((2,2)\) and is perpendicular to the line \(3x + y = 3\). Its y-intercept is
\(\dfrac{1}{3}\)
\(\dfrac{2}{3}\)
1
\(\dfrac{4}{3}\)
Question. 40
The ratio in which the line \(3x + 4y + 2 = 0\) divides the distance between the lines \(3x + 4y + 5 = 0\) and \(3x + 4y - 5 = 0\) is
1 : 2
3 : 7
2 : 3
2 : 5
Question. 41
One vertex of the equilateral triangle with centroid at the origin and one side as \(x + y - 2 = 0\) is
(-1, -1)
(2, 2)
(-2, -2)
(2, -2)
Fill in the Blanks
Question. 42
If \(a, b, c\) are in A.P., then the straight lines \(ax + by + c = 0\) will always pass through ____.
Answer:
(1, -2)
Question. 43
The line which cuts off equal intercept from the axes and passes through the point \((1, -2)\) is ____.
Answer:
x + y + 1 = 0
Question. 44
Equations of the lines through the point \((3, 2)\) and making an angle of \(45^\circ\) with the line \(x - 2y = 3\) are ____.
Answer:
3x - y - 7 = 0, x + 3y - 9 = 0
Question. 45
The points \((3, 4)\) and \((2, -6)\) are situated on the ____ of the line \(3x - 4y - 8 = 0\).
Answer:
opposite sides
Question. 46
A point moves so that square of its distance from the point \((3, -2)\) is numerically equal to its distance from the line \(5x - 12y = 3\). The equation of its locus is ____.
Answer:
13(x^2 + y^2) - 83x + 64y + 182 = 0
Question. 47
Locus of the mid-points of the portion of the line \(x \, \sin\theta + y \, \cos\theta = p\) intercepted between the axes is ____.
Answer:
4x^2 y^2 = p^2 (x^2 + y^2)
True or False
Question. 48
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
Answer:
True
Question. 49
The points A \((-2,1)\), B \((0,5)\), C \((-1,2)\) are collinear.
Answer:
False
Question. 50
Equation of the line passing through the point \((a\cos^3\theta, a\sin^3\theta)\) and perpendicular to the line \(x \sec\theta + y \csc\theta = a\) is \(x \cos\theta - y \sin\theta = a \sin 2\theta\).
Answer:
False
Question. 51
The straight line \(5x + 4y = 0\) passes through the point of intersection of the straight lines \(x + 2y - 10 = 0\) and \(2x + y + 5 = 0\).
Answer:
True
Question. 52
The vertex of an equilateral triangle is \((2,3)\) and the equation of the opposite side is \(x + y = 2\). Then the other two sides are \(y - 3 = (2 \pm \sqrt{3})(x - 2)\).
Answer:
True
Question. 53
The equation of the line joining the point \((3,5)\) to the point of intersection of the lines \(4x + y - 1 = 0\) and \(7x - 3y - 35 = 0\) is equidistant from the points \((0,0)\) and \((8,34)\).
Answer:
True
Question. 54
The line \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) moves in such a way that \(\dfrac{1}{a^2} + \dfrac{1}{b^2} = \dfrac{1}{c^2}\), where \(c\) is a constant. The locus of the foot of the perpendicular from the origin on the given line is \(x^2 + y^2 = c^2\).
Answer:
True
Question. 55
The lines \(ax + 2y + 1 = 0\), \(bx + 3y + 1 = 0\) and \(cx + 4y + 1 = 0\) are concurrent if \(a, b, c\) are in G.P.
Answer:
False
Question. 56
Line joining the points \((3,-4)\) and \((-2,6)\) is perpendicular to the line joining the points \((-3,6)\) and \((9,-18)\).
Answer:
False
Match the Following
Question. 57
Match the items in Column A with Column B using the table below.
| Column A | Column B |
|---|---|
The coordinates of the points P and Q on the line \(x + 5y = 13\) which are at a distance of 2 units from the line \(12x - 5y + 26 = 0\) | (3, 1), (-7, 11) |
The coordinates of the point on the line \(x + y = 4\), which are at a unit distance from the line \(4x + 3y - 10 = 0\) | \(\left(\dfrac{1}{3}, \dfrac{11}{3}\right), \left(\dfrac{4}{3}, \dfrac{7}{3}\right)\) |
The coordinates of the point on the line joining A(-2,5) and B(3,1) such that AP = PQ = QB | \(\left(1, \dfrac{12}{5}\right), \left(-3, \dfrac{16}{5}\right)\) |
Answer:
| Column A | Matched Item from Column B |
|---|---|
The coordinates of the points P and Q on the line \(x + 5y = 13\) which are at a distance of 2 units from the line \(12x - 5y + 26 = 0\) | \(\left(1, \dfrac{12}{5}\right), \left(-3, \dfrac{16}{5}\right)\) |
The coordinates of the point on the line \(x + y = 4\), which are at a unit distance from the line \(4x + 3y - 10 = 0\) | (3, 1), (-7, 11) |
The coordinates of the point on the line joining A(-2,5) and B(3,1) such that AP = PQ = QB | \(\left(\dfrac{1}{3}, \dfrac{11}{3}\right), \left(\dfrac{4}{3}, \dfrac{7}{3}\right)\) |
Question. 58
Match the items in Column A with Column B using the table below.
| Column A | Column B |
|---|---|
Parallel to y-axis | \(\lambda = -\dfrac{3}{4}\) |
Perpendicular to \(7x + y - 4 = 0\) | \(\lambda = \dfrac{1}{3}\) |
Passes through (1,2) | \(\lambda = \dfrac{17}{41}\) |
Parallel to x-axis | \(\lambda = 3\) |
Answer:
| Column A | Matched Item from Column B |
|---|---|
Parallel to y-axis | \(\lambda = 3\) |
Perpendicular to \(7x + y - 4 = 0\) | \(\lambda = -\dfrac{3}{4}\) |
Passes through (1,2) | \(\lambda = \dfrac{1}{3}\) |
Parallel to x-axis | \(\lambda = \dfrac{17}{41}\) |
Question. 59
Match the items in Column A with Column B using the table below.
| Column A | Column B |
|---|---|
Through the point (2,1) | 2x - y = 4 |
Perpendicular to the line \(x + 2y + 1 = 0\) | x + y - 5 = 0 |
Parallel to the line \(3x - 4y + 5 = 0\) | x - y - 1 = 0 |
Equally inclined to the axes | 3x - 4y - 1 = 0 |
Answer:
| Column A | Matched Item from Column B |
|---|---|
Through the point (2,1) | x - y - 1 = 0 |
Perpendicular to the line \(x + 2y + 1 = 0\) | 2x - y = 4 |
Parallel to the line \(3x - 4y + 5 = 0\) | 3x - 4y - 1 = 0 |
Equally inclined to the axes | x + y - 5 = 0 |