**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

m_{1} = -8p / (2 - 3p)

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

m_{2} = -p/8

Because the lines are perpendicular,

m_{1 }⋅ m_{2} = -1

Substitute.

[-8p / (2 - 3p)] ⋅ [-p/8] = -1

[(8p^{2}) / 8(2 - 3p)] = -1

p^{2} / (2 - 3p) = -1

p^{2} = -1(2 - 3p)

p^{2} = -2 + 3 p

p^{2 }- 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 m_{1} be the slope of the line joining (h, 3) and (4, 1).

Formula for slope of line joining two points is given by

m = (y_{2 }- y_{1}) / (x_{2 }- x_{1})

Then,

m_{1} = (1 - 3) / (4 - h)

m_{1} = -2 / (4 - h)

Let m_{2} be the slope of the line 7x – 9y – 19 = 0.

m_{2} = -7/(-9)

m_{2 }= 7/9

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

Then,

m_{1 }⋅ m_{2} = -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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