Find the value of 'p' if the lines, 5x – 3y + 2 = 0 and 6x – py + 7 = 0 are perpendicular to each other. Hence, find the equation of a line passing through (– 2, – 1) and parallel to 6x – py + 7 = 0.

First line:

5x − 3y + 2 = 0

3y = 5x + 2

y = (5/3)x + 2/3

Slope m₁ = 5/3

Second line:

6x − py + 7 = 0

py = 6x + 7

y = (6/p)x + 7/p

Slope m₂ = 6/p

For perpendicular lines:

m₁m₂ = −1

(5/3)(6/p) = −1

10/p = −1

p = −10

Therefore the second line is:

6x + 10y + 7 = 0

Its slope is:

−3/5

The required line passes through (−2, −1) and is parallel to it.

Using point-slope form:

y + 1 = (−3/5)(x + 2)

5y + 5 = −3x − 6

3x + 5y + 11 = 0

Answer:

p = −10 Required line: 3x + 5y + 11 = 0.

Find the slopes of the two given lines and use m₁m₂ = −1 for perpendicular lines. The second part uses the same slope as the line 6x + 10y + 7 = 0 because parallel lines have equal slopes.