ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2

More Exercises

Question 1.
State which one of the following is true : The straight lines y = 3x – 5 and 2y = 4x + 7 are
(i) parallel
(ii) perpendicular
(iii) neither parallel nor perpendicular.
Solution:
Slope of line y = 3x – 5 = 3
and slope of line 2y = 4x + 7
⇒ y = 2x + \(\\ \frac { 7 }{ 2 } \) = 2.
∴ Slope of both the lines are neither equal nor their product is – 1.
∴ These line are neither parallel nor perpendicular.

Worksheets for Class 10 Maths

Question 2.
If 6x + 5y – 7 = 0 and 2px + 5y + 1 = 0 are parallel lines, find the value of p.
Solution:
In equation
6x + 5 y – 7 = 0
⇒ 5y = -6x + 7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q2.1

Question 3.
Lines 2x – by + 5 = 0 and ax + 3y = 2 are parallel. Find the relation connecting a and b. (1991)
Solution:
In equation 2x – by + 5 = 0
⇒ – by = – 2x – 5
⇒ y = \(\\ \frac { 2 }{ b } \) + \(\\ \frac { 5 }{ b } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q3.1

Question 4.
Given that the line \(\\ \frac { y }{ 2 } \) = x – p and the line ax + 5 = 3y are parallel, find the value of a. (1992)
Solution:
In equation y = x – p
⇒ y = 2x – 2p
Slope (m1) = 2
In equation ax + 5 = 3y
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q4.1

Question 5.
If the lines y = 3x + 7 and 2y + px = 3 perpendicular to each other, find the value of p. (2006)
Solution:
Gradient m1 of the line y = 3x + 7 is 3
2y + px = 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q5.1

Question 6.
Find the value of k for which the lines kx – 5y + 4 = 0 and 4x – 2y + 5 = 0 are perpendicular to each other. (2003)
Solution:
Given
In equation, kx – 5y + 4 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q6.1

Question 7.
If the lines 3x + by + 5 = 0 and ax – 5y + 7 = 0 are perpendicular to each other, find the relation connecting a and b.
Solution:
Given
In the equation 3x + by + 5 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q7.1

Question 8.
Is the line through ( – 2, 3) and (4, 1) perpendicular to the line 3x = y + 1 ?
Does the line 3x = y + 1 bisect the join of ( – 2, 3) and (4, 1). (1993)
Solution:
Slope of the line passing through the points
(-2, 3) and (4, 1) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-x_{ 1 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q8.1
Hence the line 3x = y + 1 bisects the line joining the points (-2, 3), (4, 1).

Question 9.
The line through A ( – 2, 3) and B (4, b) is perpendicular to the line 2x – 4y = 5. Find the value of b.
Solution:
Gradient (m1) of the line passing through the
points A (-2, 3) and B (4, b)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q9.1

Question 10.
If the lines 3x + y = 4, x – ay + 7 = 0 and bx + 2y + 5 = 0 form three consecutive sides of a rectangle, find the value of a and b.
Solution:
In the line 3x + y = 4 …(i)
⇒ y = – 3x + 4
Slope (m1) = – 3
In the line x – ay + 7 = 0…..(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q10.1
⇒ -b = -6 ⇒ b = 6
Hence a = 3, b = 6

Question 11.
Find the equation of a line, which has the y-intercept 4, and is parallel to the line 2x – 3y – 7 = 0. Find the coordinates of the point where it cuts the x-axis. (1998)
Solution:
In the given line 2x – 3y – 7 = 0
⇒ 3y = 2x – 7
⇒ \(y=\frac { 2 }{ 3 } x-\frac { 7 }{ 3 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q11.1

Question 12.
Find the equation of a straight line perpendicular to the line 2x + 5y + 7 = 0 and with y-intercept – 3 units.
Solution:
In the line 2x + 5y + 7 = 0
⇒ 5y = – 2x – 7
⇒ \(y=\frac { -2 }{ 5 } x-\frac { 7 }{ 5 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q12.1

Question 13.
Find the equation of a st. line perpendicular to the line 3x – 4y + 12 = 0 and having same y-intercept as 2x – y + 5 = 0.
Solution:
In the given line 3x – 4y + 12 = 0
⇒ 4y = 3x + 12
⇒ y = \(\\ \frac { 3 }{ 4 } x+3\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q13.1

Question 14.
Find the equation of the line which is parallel to 3x – 2y = – 4 and passes through the point (0, 3). (1990)
Solution:
In the given line 3x – 2y = – 4
⇒ 2y = 3x + 4
⇒ y = \(\\ \frac { 3 }{ 2 } x+2\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q14.1

Question 15.
Find the equation of the line passing through (0, 4) and parallel to the line 3x + 5y + 15 = 0. (1999)
Solution:
In the given equation 3x + 5y + 15 = 0
⇒ 5y = – 3x – 15
⇒ y = \(\\ \frac { -3 }{ 5 } x-3\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q15.1

Question 16.
The equation of a line is y = 3x – 5. Write down the slope of this line and the intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to the line and which passes through the point (0, 5).
Solution:
In the given line y = 3x – 5
Here slope (m1) = 3
Substituting x = 0, then y = – 5
y-intercept = – 5
The slope of the line parallel to the given line
will be 3 and passes through the point (0, 5).
Equation of the line will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q16.1

Question 17.
Write down the equation of the line perpendicular to 3x + 8y = 12 and passing through the point ( – 1, – 2).
Solution:
In the given line 3x + 8y = 12
⇒ 8y = -3x + 12
⇒ \(y=\frac { -3 }{ 8 } x+\frac { 12 }{ 8 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q17.1

Question 18.
(i) The line 4x – 3y + 12 = 0 meets the x-axis at A. Write down the co-ordinates of A.
(ii) Determine the equation of the line passing through A and perpendicular to 4x – 3y + 12 = 0. (1993)
Solution:
(i) In the line 4x – 3y + 12 = 0 …(i)
3y = 4x + 12
⇒ y = \(\\ \frac { 4 }{ 3 } x+4\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q18.1

Question 19.
Find the equation of the line that is parallel to 2x + 5y – 7 = 0 and passes through the mid-point of the line segment joining the points (2, 7) and ( – 4, 1).
Solution:
The given line 2x + 5y – 7 = 0
5y = -2x + 7
⇒ \(y=\frac { -2 }{ 5 } x+\frac { 7 }{ 5 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q19.1

Question 20.
Find the equation of the line that is perpendicular to 3x + 2y – 8 = 0 and passes through the mid-point of the line segment joining the points (5, – 2), (2, 2).
Solution:
In the given line 3x + 2y – 8=0
⇒ 2y = – 3x + 8
⇒ y = \(\\ \frac { -3 }{ 2 } x+4\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q20.1

Question 21.
Find the equation of a straight line passing through the intersection of 2x + 5y – 4 = 0 with x-axis and parallel to the line 3x – 7y + 8 = 0.
Solution:
Let the point of intersection of the line 2x + 5y – 4 = 0 and x-axis be (x, 0)
Substituting the value of y in the equation
2x + 5 × 0 – 4 = 0
⇒ 2x – 4 = 0
⇒ 2x = 4
⇒ x = \(\\ \frac { 4 }{ 2 } \) = 2
Coordinates of the points of intersection will be (2, 0)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q21.1

Question 22.
The equation of a line is 3x + 4y – 7 = 0. Find
(i) the slope of the line. .
(ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y – 10 = 0. (2010)
Solution:
(i) Equation of the line is 3x + 4y – 1 = 0
⇒ 4y = 7 – 3x
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q22.2

Question 23.
Find the equation of the line perpendicular from the point (1, – 2) on the line 4x – 3y – 5 = 0. Also find the co-ordinates of the foot of perpendicular.
Solution:
In the equation 4x – 3y – 5 = 0,
⇒ 3y = 4x – 5
⇒ \(y=\frac { 4 }{ 3 } x-\frac { 5 }{ 3 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q23.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q23.2

Question 24.
Prove that the line through (0, 0) and (2, 3) is parallel to the line through (2, – 2) and (6, 4).
Solution:
Given that
Slope of the line through (0, 0) and (2, 3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q24.1

Question 25.
Prove that the line through,( – 2, 6) and (4, 8) is perpendicular to the line through (8, 12) and (4, 24).
Solution:
Given that
Slope of the line through (-2, 6) and (4, 8)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q25.1

Question 26.
Show that the triangle formed by the points A (1, 3), B (3, – 1) and C ( – 5, – 5) is a right angled triangle by using slopes.
Solution:
Slope (m1) of line by joining the points
A(1, 3), B (3, -1) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ x_{ 2 }-{ x }_{ 1 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q26.1

Question 27.
Find the equation of the line through the point ( – 1, 3) and parallel to the line joining the points (0, – 2) and (4, 5).
Solution:
Slope of the line joining the points (0, -2) and (4, 5) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ x_{ 2 }-{ x }_{ 1 } } \)
= \(\frac { 5+2 }{ 4-0 } =\frac { 7 }{ 4 } \)
Slope of the line parallel to it passing through (-1, 3) = \(\\ \frac { 7 }{ 4 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q27.1

Question 28.
A ( – 1, 3), B (4, 2), C (3, – 2) are the vertices of a triangle.
(i) Find the coordinates of the centroid G of the triangle.
(ii) Find the equation of the line through G and parallel to AC
Solution:
Given, A (-1, 3), B (4, 2), C (3, -2)
(i) Coordinates of centroid G
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q28.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q28.2

Question 29.
The line through P (5, 3) intersects y-axis at Q.
(i) Write the slope of the line.
(ii) Write the equation of the line.
(iii) Find the coordinates of Q.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q29.1
Solution:
(i) Here θ = 45°
So, slope of the line = tan θ = tan 45° = 1
(ii) Equation of the line through P and Q is
y – 3 = 1(x – 5) ⇒ y – x + 2 = 0
(iii) Let the coordinates of Q be (0, y)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q29.2

Question 30.
In the adjoining diagram, write down
(i) the co-ordinates of the points A, B and C.
(ii) the equation of the line through A parallel to BC. (2005)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q30.1
Solution:
From the given figure, it is clear that
co-ordinates of A are (2, 3) of B are ( -1, 2)
and of C are (3, 0).
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q30.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q30.3

Question 31.
Find the equation of the line through (0, – 3) and perpendicular to the line joining the points ( – 3, 2) and (9, 1).
Solution:
The slope (m1) of the line joining the points (-3, 2) and (9, 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q31.1

Question 32.
The vertices of a triangle are A (10, 4), B (4, – 9) and C ( – 2, – 1). Find the equation of the altitude through A. The perpendicular drawn from a vertex of a triangle to the opposite side is called altitude.
Solution:
Vertices of ∆ABC are A (10, 4), B (4, -9) and C( – 2, – 1)
Slope of the line BC (m1) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q32.1

Question 33.
A (2, – 4), B (3, 3) and C ( – 1, 5) are the vertices of triangle ABC. Find the equation of :
(i) the median of the triangle through A
(ii) the altitude of the triangle through B
Solution:
(i) D is the mid-point of BC
Co-ordinates of D will
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q33.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q33.2

Question 34.
Find the equation of the right bisector of the line segment joining the points (1, 2) and (5, – 6).
Solution:
Slope of the line joining the points (1, 2) and (5, -6)
\({ m }_{ 1 }=\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } =\frac { -6-2 }{ 5-1 } =\frac { -8 }{ 4 } =-2\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q34.1

Question 35.
Points A and B have coordinates (7, – 3) and (1, 9) respectively. Find
(i) the slope of AB.
(ii) the equation of the perpendicular bisector of the line segment AB.
(iii) the value of ‘p’ if ( – 2, p) lies on it.
Solution:
Coordinates of A are (7, -3), of B = (1, 9)
(i) ∴ slope (m)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q35.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q35.2

Question 36.
The points B (1, 3) and D (6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal AC.
Solution:
Slope of BD (m1) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \)
= \(\frac { 8-3 }{ 6-1 } =\frac { 5 }{ 5 } =1\)
Diagonal AC is perpendicular bisector of diagonal BD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q36.1

Question 37.
ABCD is a rhombus. The co-ordinates of A and C are (3, 6) and ( – 1, 2) respectively. Write down the equation of BD. (2000)
Solution:
Co-ordinates of A (3, 6), C (-1, 2)
Slope of AC (m1) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \)
= \(\frac { 2-6 }{ -1-3 } =\frac { -4 }{ -4 } =1\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q37.1

Question 38.
Find the equation of the line passing through the intersection of the lines 4x + 3y = 1 and 5x + 4y = 2 and
(i) parallel to the line x + 2y – 5 = 0
(ii) perpendicular to the x-axis.
Solution:
4x + 3y = 1 …(i)
5x + 4y = 2 …(ii)
Multiplying (i) by 4 and (ii) by 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q38.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q38.2

Question 39.
(i) Write down the co-ordinates of the point P that divides the line joining A ( – 4, 1) and B (17, 10) in the ratio 1 : 2.
(ii) Calculate the distance OP where 0 is the origin
(iii) In what ratio does the y-axis divide the line AB?
Solution:
(i) Co-ordinate A (-4, 1) and B (17, 10) P divides it in the ratio of 1 : 2
Let the co-ordinates of P will be (x, y)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q39.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q39.2

Question 40.
Find the image of the point (1, 2) in the line x – 2y – 7 = 0
Solution:
Draw a perpendicular from the point P(1, 2) on the line, x – 2y – 7 = 0
Let P’ is the image of P and let its
co-ordinates sue (α, β) slope of line x – 2y – 7 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q40.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q40.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q40.3

Question 41.
If the line x – 4y – 6 = 0 is the perpendicular bisector of the line segment PQ and the co-ordinates of P are (1, 3), find the co-ordinates of Q.
Solution:
Let the co-ordinates of Q be (α, β) and let the line x – 4y – 6 = 0 is the
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q41.1
perpendicular bisector of PQ and it intersects the line at M.
M is the mid point of PQ Now slope of line x – 4y – 6 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q41.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q41.3

Question 42.
OABC is a square, O is the origin and the points A and B are (3, 0) and (p, q). If OABC lies in the first quadrant, find the values of p and q. Also write down the equations of AB and BC.
Solution:
OA = \(\sqrt { { (3-0) }^{ 2 }+{ (0-0) }^{ 2 } } \)
\(\sqrt { { (3-0) }^{ 2 }+{ (0) }^{ 2 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q42.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2 Q42.2

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