Chapter 13 Equation of a Straight Line Class 10 ICSE Mathematics

Exercise 13.1
1. Find is the following points lie on the given line or not:
(i) (1, 3) on the line 2x + 3y = 11
(ii) (5, 3) on the line 3x - 5y + 5 = 0
(iii) (2, 4) on the line y = 2x - 1
(iv) (-1, 5) on the line 3x = 2y - 13
(iv) (-1, 5) on the line 3x = 2y - 13
Answer
(i) (1, 3) on the line 2x + 3y = 11
From the question it is given that,
Point = (1, 3)
Line = 2x + 3y = 11
Now, put x = 1 and y = 3
Consider Left Hand Side (LHS) = 2x + 3y
= 2(1) + 3(3)
= 2 + 9
= 11
Right Hand Side (RHS) = 11
By comparing LHS and RHS
LHS = RHS
⇒ 11 = 11
Therefore, point lie on the given line.
(ii) (5, 3) on the line 3x - 5y + 5 = 0
From the question it is given that,
Point = (5, 3)
Line = 3x - 5y + 5 = 0
Now, put x = 5 and y = 3
Consider Left Hand Side (LHS) = 3x - 5y + 5
= 3(5) - 5(3) + 5
= 15 - 15 + 5
= 5
Right Hand Side (RHS) = 0
By comparing LHS and RHS
LHS ≠ RHS
⇒ 5 ≠ 0
Therefore, point does not lie on the given line.
(iii) (2, 4) on the line y = 2x - 1
From the question it is given that,
Point = (2, 4)
Line = y = 2x - 1
Now, put x = 2 and y = 4
Consider Left Hand Side (LHS) = 4
Right Hand Side (RHS) = 2x - 1
= 2(2) - 1
= 4 - 1
= 3
By comparing LHS and RHS
LHS ≠ RHS
4 ≠ 3
Therefore, point does not lie on the given line.
(iv) (-1, 5) on the line 3x = 2y - 13
From the question it is given that,
Point = (-1, 5)
Line = 3x = 2y - 15
Now, put x = -1 and y = 5
Consider Left Hand Side (LHS) = 3x
= 3(-1)
= - 3
Right Hand Side (RHS) = 2y - 13
= 2(5) - 15
= 10 - 13
= -3
By comparing LHS and RHS
LHS = RHS
⇒ - 3 = - 3
Therefore, point lie on the given line.
(v) (7, -2) on the line 5x + 7y = 11
From the question it is given that,
Point = (7, -2)
Line = 5x + 7y = 11
Now, put x = 7 and y = -2
Consider Left Hand Side (LHS) = 5x + 7y
= 5(7) +7(-2)
= 35 - 14
= 21
Right Hand Side (RHS) = 11
By comparing LHS and RHS
LHS ≠ RHS
⇒ 21 ≠11
Therefore, point does not lie on the given line.
2. Find the value of m if the line 2x + 5y + 12 = 0 passes through the point (4, m)
Answer
From the question it is given that,
The line 2x + 5y + 12 = 0 passes through the point (4, m)
We have to find the value of m,
So, put x = 4 and y = m
2x + 5y + 12 = 0
⇒ 2(4) + 5(m) + 12 = 0
⇒ 8 + 5m + 12 = 0
⇒ 5m + 20 = 0
⇒ 5m = - 20
⇒ m = -20/5
⇒ m = -4
Therefore, the value of m is -4.
3. Find the value of P if the line 3y = 5x - 7 passes through the point (p, 6).
Answer
From the question it is given that,
The line 3y = 5x - 7 passes through the point (p, 6)
We have to find the value of p,
So, put x = p and y = 6
3y = 5x - 7
⇒ 3(6) = 5(P) - 7
⇒ 18 = 5P - 7
⇒ 18 + 7 = 5P
⇒ 25 = 5P
⇒ P = 25/5
⇒ P = 5
Therefore, the value of P is 5.





4. Find the value of a if the line 4x = 11 - 3y passes through the point (a, 5).



Answer

From the question it is given that,

The line 4x = 11 - 3y passes through the point (a, 5)

We have to find the value of a,

So, put x = a and y = 5

4x = 11 - 3y

⇒ 4
(a) = 11 - 3(5)

⇒ 4a = 11 - 15

⇒ 4a = - 4

⇒ a = -4/4

⇒ a = - 1

Therefore, the value of a is -1.





5. The line y = 6 - 3x/2 passes through the point (r, 3). Find the value of r.



Answer

From the question it is given that,

The line y = 6 - 3x/2 passes through the point (r, 3)

We have to find the value of r,

So, put x = r and y = 3

y = 6 - 3x/2

⇒ 3 = 6 - (3(r))/2

⇒ 3 = (12 - 3r)/2

⇒ 6 = 12 - 3r

⇒ 3r = 12 - 6

⇒ 3r = 6

⇒ r = 6/3

⇒ r = 2

Therefore, the value of r is 2.





6. The line (3 + 5y)/2 = (4x - 7)/3 passes through the point (1, k). find the value of k



Answer

From the question it is given that,

The line (3 + 5y)/2 = (4x - 7)/3 passes through the point (1, k)

We have to find the value of k,

So, put x = 1 and y = k

(3 + 5y)/2 = (4x - 7)/3

⇒ (3 + 5(k))/2 = (4(1) - 7)/3

⇒ 3(3 + 5k) = 2(4 - 7)

⇒ 9 + 15k = 2(- 3)

⇒ 9 + 15k = - 6

⇒ 15k = - 6 - 9

⇒ 15k = - 15

⇒ k = -15/15

⇒ k = - 1

Therefore, the value of k is - 1.





7. The line 4x + 3y = 11 bisects the join of (6, m) and (4, 9). Find the value of m.



Answer

Let us assume the point of intersection of CD and line 4x + 4y = 11 be the point Q(a, b)

From the question it is given that, line 4x + 3y = 11 bisects the line segment CD,

So, CQ: QD = 1: 1

Then, the coordinates of Q are,

Q(a, b) = Q[((6 + 4)/2), ((m - 9)/2)]

= Q[5, ((m - 9)/2)]

Since Q(a, b) lies on the line 4x + 3y = 11,

Where, x = 5, y = (m - 9)/2

4(5) + 3((m - 9)/2) = 11

⇒ 20 + (3m - 27)/2 = 11

⇒ 40 + 3m - 27 = 22

⇒ 3m + 13 = 22

⇒ 3m = 22 - 13

⇒ 3m = 9

⇒ m = 9/3

⇒ m = 3

Therefore, value of m is 3.





8. The line 2x - 5y + 31 = 0 bisects the join of (-4, 5) and (p, 9). Find the value of p.



Answer

Let us assume the point of intersection of CD and line 4x + 4y = 11 be the point Q(a, b)

From the question it is given that, line 2x - 5y + 31 = 0 bisects the line segment CD,

So, CQ: QD = 1: 1

Then, the coordinates of Q are,

Q(a, b) = Q[((-4 + P)/2), ((5 + 9)/2)]

= Q[((-4 + P)/2), 7]

Since Q(a, b) lies on the line 2x - 5y + 31 = 0,

Where, x = (-4 + P)/2, y = 7

2((-4 + P)/2) - 5(7) + 31 = 0

⇒ (-8 + 2P)/2 - 35 + 31 = 0

⇒ (-8 + 2P)/2 - 4 = 0

⇒ -8 + 2P - 8 = 0

⇒ -16 + 2P = 0

⇒ 2P = 16

⇒ P = 16/2

⇒ P = 8

Therefore, value of P is 8.





9. The line segment formed by the points (3, 7) and (-7, Z) is bisected by the line 3x + 4y = 18. Find the value of z.



Answer

Let us assume the point of intersection of CD and line 3x + 4y = 18 be the point Q(a, b)

From the question it is given that, line 3x + 4y = 18 bisects the line segment CD,

So, CQ: QD = 1: 1

Then, the coordinates of Q are,

Q(a, b) = Q[((-3 + 7)/2), ((7 + z)/2)]

= Q[-2, ((7 + z)/2)]

Since Q(a, b) lies on the line 3x + 4y = 18,

Where, x = - 2, y = (7 + z)/2

3x + 4y = 18

⇒ 3(-2) + 4((7 + z)/2) = 18

⇒ -6 + (28 + 4z)/2 = 18

⇒ -12 + 28 + 4z = 36

⇒ 16 + 4z = 36

⇒ 4z = 36 - 16

⇒ 4z = 20

⇒ z = 20/4

⇒ z = 5
Therefore, value of z is 5.





10. The line 5x - 3y + 1 = 0 divides the join of (2, m) and (7, 9) in the ratio 2: 3. Find the value of m.



Answer

Let us assume the point of intersection of CD and line 5x - 3y + 1 = 0be the point Q(a, b)

From the question it is given that, line 5x - 3y + 1 = 0divides the line segment CD are in the ratio 2: 3,

So, CQ: QD = 2: 3

So, Point C become 3(2, m) = (6, 3m)

D become 2(7, 9) = (14, 18)

Then, the coordinates of Q are,

Q(a, b) = Q[((14 + 6)/5), ((18 + 3m)/5)]

= Q[4, ((18 + 3m)/5)]

Since Q(a, b) lies on the line 5x - y + 1 = 0,

Where, x = 4, y = (18 + 3m)/5

5x - 3y + 1 = 0

⇒ 5(4) - 3((18 + 3m)/5) + 1 = 0

⇒ 20 - (54 + 9m)/5 + 1 = 0

⇒ 21 - (54 + 9m)/5 = 0

⇒ 105 - 54 - 9m = 0

⇒ 51 - 9m = 0

⇒ 9m = 51

⇒ m = 51/9

⇒ m = 17/3 ...[divide both by 3]

Therefore, value of m is 17/3.





11.
The line 7x - 8y = 4 divides the join of (-8, -4) and (6, k) in the ratio 2: 5. Find the value of k.



Answer

Let us assume the point of intersection of CD and line 7x - 8y = 4be the point Q(a, b)

From the question it is given that, line 7x - 8y = 4divides the line segment CD are in the ratio 2: 5,

So, CQ: QD = 2: 5

So, Point C become 5(-8, -4) = (-40, -20)

D become 2(6, k) = (12, 2k)

Then, the coordinates of Q are,

Q(a, b) = Q[((12 - 40)/7), ((2k - 20)/7)]

= Q[-4, ((2k - 20)/7)]

Since Q(a, b) lies on the line 7x - 8y = 4,

Where, x = - 4, y = (2k - 20)/7

7(-4) - 8((2k - 20)/7) = 4

⇒ - 28 - (16k - 160)/7 = 4

⇒ - 196 - 16k + 160 = 28

⇒ - 36 - 16k = 28

⇒ 16k = - 36 - 28

⇒ 16k = - 64

⇒ k = -64/16

⇒ k = -4

Therefore, value of k is - 4.





12.
The line 5x + 3y = 25 divides the join of (b, 4) and (5, 8) in the ratio 1: 3. Find the value of b.



Answer

Let us assume the point of intersection of CD and line 5x + 3y = 25be the point Q(a, b)

From the question it is given that, line 5x + 3y = 25divides the line segment CD are in the ratio 1: 3,

So, CQ: QD = 1: 3

So, Point C become 3(b, 4) = (3b, 12)

D become 1(5, 8) = (5, 8)

Then, the coordinates of Q are,

Q(a, b) = Q[((5 + 3b)/4), ((8 + 12)/4)]

= Q[((5 + 3b)/4), 5]

Since Q(a, b) lies on the line 5x + 3y = 25,

Where, x = (5 + 3b)/4, y = 5

5((5 + 3b)/4) + 3(5) = 25

⇒ (25 + 15b)/4 + 15 = 25

⇒ 25 + 15b + 60 = 100

⇒ 15b + 85 = 100

⇒ 15b = 100 - 85

⇒ 15b = 15

⇒ b = 15/15

⇒ b = 1

Therefore, value of b is 1.





13.
P is a point on the line segment AB dividing it in the ratio 2: 3. If the coordinates of A and B are (-2, 3) and (8, 8), find if P lies on the line 7x - 2y = 4.



Answer

From the question it is given that,

The coordinates of A and B are (-2, 3) and (8, 8)

The line segment AB dividing it in the ratio 2: 3

So, AP: PB = 2: 3

Then, A = 3(-2, 3) = (-6, 9)

B = 2(8, 8) = (16, 16)

Then, the coordinates of P are,

P(a, b) = P[((16 - 6)/5), ((16 + 9)/5)]

= P[2, 5]

Since P(a, b) lies on the line 7x - 2y = 4,

Where, x = 2, y = 5

Consider Left Hand Side (LHS) = 7x - 2y

= 7(2) - 2(5)

= 14 - 10

= 4

Right Hand Side (RHS) = 4

By comparing LHS and RHS

LHS = RHS

4 = 4

Therefore, point P(2, 5) lie on the given line 7x - 2y = 4.





14.
L is a point on the line segment PQ dividing it in the ratio 1: 3. If the coordinates of P and Q are (3, 7) and (11, -5) respectively, find if L lies on the line 2x + 5y = 20.



Answer

From the question it is given that,

The coordinates of P and Q are (3, 7) and (11, -5) respectively

The line segment PQ dividing it in the ratio 1: 3

So, LP: LQ = 1: 3

Then, P = 3(3, 7) = (9, 21)

Q = 1(11, -5) = (11, -5)

Then, the coordinates of L are,

L(a, b) = L[((11 + 9)/4), ((- 5 + 21)/4)]

= L[5, 4]

Since L(a, b) lies on the line 2x + 5y = 20,

Where, x = 5, y = 4

Consider Left Hand Side (LHS) = 2x + 5y

= 2(5) + 5(4)

= 10 + 20

= 30

Right Hand Side (RHS) = 20

By comparing LHS and RHS

LHS ≠ RHS

30 ≠ 20

Therefore, point L(a, b) does not lie on the given line 2x + 5y = 20.





15.
The line segment formed by two points A(2, 3) and B(5, 6) is divided by a point in the ratio 1: 2. Find, whether the point of intersection lies on the line 3x - 4y + 5 = 0.



Answer

From the question it is given that,

The coordinates of A(2, 3) and B(5, 6).

The line segment AB dividing it in the ratio 1: 2

So, AL: LB = 1: 3

Then, A = 2(2, 3) = (4, 6)

B = 1(5, 6) = (5, 6)

Then, the coordinates of L are,

L(a, b) = L[((5 + 4)/3), ((6 + 6)/3)]

= L[3, 4]

Since L(a, b) lies on the line 3x - 4y + 5 = 0,

Where, x = 3, y = 4

Consider Left Hand Side (LHS) = 3x - 4y + 5

= 3(3) - 4(4) + 5

= 9 - 16 + 5

= -2

Right Hand Side (RHS) = 0

By comparing LHS and RHS

LHS ≠ RHS

-2 ≠ 0

Therefore, point L (a, b) does not lie on the given line 3x - 4y + 5 = 0.





Exercise 13.2

1. Find the slope of a line, correct of two decimals, whose inclination is



(a) 60°



(b) 50°



(c) 45°



(d) 75°



(e) 30°
Answer





2. Find the inclination of a line whose gradient is
(a) 0.4663
(b) 1.4281
(c) 3.0777
(d) 5.6713
(e) 0.5317
Answer










3. Find the slope of a line passing through the given pair of points.
(a) (2, 5) and (-1, 8)
(b) (3, 7) and (5, 13)
(c) (-5, - 1) and (-9, - 7)
(d) (9, - 2) and (-5, 5)
(e) (0, 5) and (5, 0)
Answer





4. Find the slope of a line passing through the following pairs of points:
(a)2m
²
, 2am) and (
p²
m
²
, 2pm)
(b) (5pq,
p²
q) and (5qr, qr
²
)
Answer







5. Find the slope of a line parallel to the given lines:
(a) 3x - 2y = 5
(b) x + 3y = 7
(c) 5x - y = 10
(d) 4x - 2y = 3
(e) 5x + 2y = 11
Answer







6. Find the value of a line perpendicular to the given lines:
(a) 2x - 3y = 4
(b) 5x + 2y - 9 = 0
(c) 3x + 4y = 13
(d) x - 4y = 8
(e) 9x - 3y = 5
Answer





7. Find the slope of a line parallel to AB, if the coordinates of A and B are (3, - 1) and (-7, 5) respectively.
Answer







8. Find the slope of a line parallel to MN, if the coordinates of M and N are (4, 9) and (-2, 3) respectively.
Answer







9. Find the slope of a line parallel to PQ, if the coordinates of P and Q are (-11, 3) and (7, 13) respectively.
Answer
Slope of line PQ = (y₂ - y₁)/(x₂ - x₁)
= (13 + 3)/(7 - 11)
= 16/-4
= -4
Slope of line parallel to PQ = Slope of PQ
= -4
10. Find x if the slope of the line passing through the points (x, 9) and (12, 6) is -1/3.
Answer
Slope of line AB
⇒ -1/3 = (6 - 9)/(12 - x)
⇒ x - 12 = - 9
⇒ x = 3
11. Find m if the slope of the line passing through the point (-7, 5) and (2, m) is 1/3.
Answer


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12. Find p if the slope of the line passing through the point (-2, 5) and (p, 2p + 1) is 1.
Answer







13. Find the slope and the inclination of the line passing through (8, 1) and (6, 5).
Answer







14. Find the slope and inclination of the line passing through (- 5, 7) and (7, - 5).
Answer







15. Find the value of a line parallel to the following lines:
(a) x = y/2 - 5
(b) x = 3y/2 + 2
(c) 3x/4 + 5y/2 = 7
(d) x/4 + y/3 = 1
(e) 2x/5 + y/3 = 2
Answer





16. Find the slope of a line perpendicular to the following lines:
(a) x/2 + y/3 = 4/3
(b) x - 3y/2 + 1 = 0
(c) 3x/4 - y = 5
(d) 3x - 5y = 9
(e) 4x + y = 7
Answer







17. Without using distance formula, show that the points A(12, 8), B (- 2, 6) and C (6, 0) from a right-angled triangle.
Answer







18. Without distance formula, show that the points P (2, 1), Q (- 5, - 1), R (1, 5) and S (- 2, - 1) from a parallelogram.
Answer







19. Without distance formula, show that the points A (5, 8), B (4, 4), C(0, 0) and D(1, 9) form a rhombus.
Answer







20. Without distance formula, show that the points M (1, 7), N (4, 8), O (5, 5) and P (2, 4) form a square.
Answer





Exercise 13.3
1. Find the slope and the y-intercept of the follow lines:
(a) 5x - 2y = 6
(b) 3x + y = 7
(c) 4y = 5x - 8
(d) 2x + 3y = 12
(e) x - 2 = (5 - 3y)/2
Answer







2. Find the equation of a line whose slope and y-intercept are
(a) m = -6/5, c = 3
(b) m = 2/3, c = 2
(c) m = -1/2, c = 5
(d) m = - 3, c = - 1
(e) m = 2, c = - 5
Answer

Equation of line with slope and y-intercept



3. Find the equation of a line passing through (2, 5) and making an angle of 30°with the positive direction of the x-axis.
Answer





4. Find the equation of a line passing through (3, 7) and making an angle of 60° with the negative direction of the x-axis.
Answer










5. Find the equation of a line passing through (8, 3) and making an angle of 45°with the positive direction of the y-axis.
Answer







6. Find the equation of a line passing through (2, 9) and parallel to the line 3x + 4y = 11.
Answer







7. Find the equation of a line passing through (-5, - 1) and perpendicular to the line 3x + y = 9.
Answer







8. Find the equation of the perpendicular bisector of AB if the coordinates of A and B are (-2, 6) and (4, - 6).
Answer







9. Find the equation of a line perpendicular to the join of A (3, 5) and B(-1, 7) if it passes through the midpoint of AB.
Answer







10. Find the equation of a line passing through the intersection of x + 3y = 6 and 2x - 3y = 12 and parallel to the line 5x + 2y = 10
Answer







11. Find the equation of a line passing through the intersection of x + 2y + 1 = 0 and 2x - 3y = 12 and perpendicular to the line 2x + 3y = 9.
Answer







12. Find the equation of a line passing through the intersection of x/10 + y/5 = 14 and x/8 + y/6 = 15 and perpendicular to the line x - 2y = 5.
Answer







13. The lines px + 5y + 7 = 0 and 2y = 5x - 6 are perpendicular to each other. Find p.
Answer







14. The lines 3x - 2y + 4 = 0 and 3x + my + 6 = - are parallel to each other. Find m.
Answer







15. Find the relation connecting p and q, if the lines py = 2x + 5 and qx + 3y = 2 are parallel to each other.
Answer







16. Find the relation connecting a and b, if the lines ay = 2x + 4 and 4y + bx = 2 are perpendicular to each other.
Answer







17. P (5, 3), Q (-4, 7) and R (8, 3) are the vertices of a triangle. Find the equation of the median of the triangle from p.
Answer







18. A (8, 5), B (-2, 1) and C (5, 4) are the vertices of a triangle. Find the equation of the median of the triangle through C.
Answer







19. ABCD is a rhombus. The coordinates of A and C (3, 7) and (9, 15). Find the equation of BD.



Answer







20. ABCD is a square. The coordinates of B and D are (-3, 7) and (5, - 1) respectively. Find the equation of AC.
Answer





21. The coordinates of two points P and Q are (0, 4) and (3, 7) respectively. Find:
(i) the gradient of PQ.
(ii) the equation of PQ.
(iii) the coordinates of the point where the line AB intersects the x-axis.
Answer





22. X(4, 9), Y(-5, 4) and Z(7, - 4) are the vertices of a triangle. Find the equation of the altitude of the triangle through X.
Answer