FIND THE MISSING VALUE WITH PERPENDICULAR LINES

Example 1 :

Find the values of p for which the following two straight lines are perpendicular to each other.

8px + (2 - 3p)y + 1  =  0

px + 8y + 7  =  0

Solution :

If two lines are perpendicular then product of their slopes will be equal to -1.

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

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

m₁  =  -8p / (2 - 3p)

Slope of the second line px + 8y + 7 = 0 is

m₂  =  -p/8

Because the lines are perpendicular, 

m₁ ⋅ m₂  =  -1

Substitute. 

[-8p / (2 - 3p)] ⋅ [-p/8]  =  -1

[(8p²) / 8(2 - 3p)]  =  -1

p² / (2 - 3p)  =  -1

p²  =  -1(2 - 3p)

p²  =  -2 + 3 p

p² - 3p + 2  =  0

(p - 2) (p - 1)  =  0

p - 2  =  0     (or)     p - 1  =  0

  p  =  2   (or)   p  =  1

Example 2 :

If the straight lines passing through the points (h, 3) and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle, then find the value of h.

Solution :

The required straight lines passing through the points (h, 3) and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle

Because the line joining the points (h, 3) and (4, 1) and the line 7x - 9y - 19  =  0 are perpendicular the product of their slopes will be equal to -1

Let m₁ be the slope of the line joining (h, 3) and (4, 1). 

Formula for slope of line joining two points is given by 

m  =  (y₂ - y₁) / (x₂ - x₁)

Then, 

m₁  =  (1 - 3) / (4 - h)

m₁  =  -2 / (4 - h)

Let m₂ be the slope of the line 7x – 9y – 19 = 0.

m₂  =  -7/(-9)

m₂  =  7/9

Because the lines intersect at right angle, they are perpendicular. 

Then, 

m₁ ⋅ m₂  =  -1

[-2 / (4 - h)] x (7 / 9)  =  -1

- 14 / 9(4 - h)  =  -1

Multiply each side by (-1).

- 14 / 9(4 - h)  =  -1

14 / (36 - 9h)  =  1

14  =  36 - 9h

Add 9h to each side. 

9h + 14  =  36

Subtract 14 from each side. 

h  =  22/9

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