WORD PROBLEMS INVOLVING PARALLEL AND PERPENDICULAR LINES WORKSHEET

Question 1 :

If the straight lines

y/2 = x – p and ax + 5 = 3y

are parallel, then find a.

Question 2 :

Find the value of a if the straight lines

5x – 2y – 9 = 0 and ay + 2x – 11 = 0

are perpendicular to each other.

Question 3 :

Find the value of p for which the straight lines

8px + (2 - 3p)y + 1 = 0 and px + 8y – 7 = 0

are perpendicular to each other.

Question 4 :

If the straight line passing through the points

(h, 3) and (4, 1)

intersects the line

7x – 9y – 19 = 0

at a right angle, find the value of h.

1. Answer :

y/2 = x – p and ax + 5 = 3y

Since the given lines are parallel, they will have same slope.

Slope of the line y/2 = x – p :

y/2 = x – p

y = 2(x - p)

y = 2x – 2p

Comparing y = 2x – 2p and y = mx + b, we get

m = 2

Slope of the 1^st line (m₁) = 2  ----(1)

Slope of the line 3 y = ax + 5 :

3y = ax + 5

y = (ax + 5)/3

y = (a/3)x + (5/3)

Slope of the 2^nd line (m₂) = a/3  ----(2)

(1) = (2)

2 = a/3

a = 6

2. Answer :

5x – 2y – 9 = 0 and ay + 2x – 11 = 0

If two lines are perpendicular then,

slope of the 1^st line x slope of the 2^nd line = -1

Slope of the line 5x – 2y – 9 = 0 :

5x – 2y – 9 = 0

slope (m) = -coefficient of x/coefficient of y

m₁ = -5/(-2)

m₁ = 5/2 ----(1)

Slope of the line ay + 2x – 11 = 0 :

2x + ay - 11 = 0

m₂ = -2/a ----(2)

Product of slopes of two lines = -1

(5/2) x (-a/2) = -1

-5a/4 = -1

-5a = -4

a = -4/(-5)

a = 4/5

3. Answer :

8px + (2 - 3p)y + 1 = 0 and px + 8y – 7 = 0

If two lines are perpendicular then,

slope of the 1^st line x slope of the 2^nd line = -1

Slope of the line 8px+(2-3p)y+1 = 0 :

m₁ = -8p/(2 - 3p) ----(1)

Slope of the line px + 8y – 7 = 0 :

m₂ = -p/8 ----(2)

m₁ x m₂ = -1

-8p/(2 - 3p) x (-p/8) = -1

p²/(2 - 3p) = -1

p² = -1(2 - 3p)

p² = -2 + 3p

p² - 3p + 2 = 0

(p - 1)(p - 2) = 0

p – 1 = 0 or p – 2 = 0

p = 1 or p = 2

4. Answer :

Here the straight which is passing through the points (h, 3) and (4, 1) is perpendicular to the line 7x – 9y – 19 = 0.

Slope of the line when two points are given :

m = (y₂ - y₁)/(x₂ - x₁)

m₁ = (1 - 3)/(4 - h)

m₁ = -2/(4 - h) ----(1)

m₂ = -7/(-9)

m₂ = 7/9 ----(2)

[-2/(4 - h)] x (7/9) = -1

[2/(4 - h)] x [7/9] = 1

14/9(4 - h) = 1

14 = 36 – 9h

9h = 36 – 14

9h = 22

h = 22/9

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