Question

Find the value of p,if the line 2x+3y+4=0 and px+2y-1=0 are perpendicular to each other ​

Answer 1

Suppose we are given a line Ax + By + C = 0 and we have to find the equation of another line which is perpendicular to this one.

Here the slope of the line Ax + By + C = 0 is,

m = - A / B

We know that the product of slopes of two perpendicular lines is always equal to -1. Let the slope of the line perpendicular to Ax + By + C = 0 be m'. So we have,

m · m' = -1

- A m' / B = -1

m' = B / A

m' = - (- B / A)

So the line perpendicular to Ax + By + C = 0 should be in the form k(Bx - Ay + D) = 0, where k is some constant and D is the term corresponding to the intercepts of this line.

If Ax + By + C = 0 is compared with 2x + 3y + 4 = 0,

A = 2 → (1)

B = 3 → (2)

If k(Bx - Ay + D) = 0 is compared with px + 2y - 1 = 0,

kB = p → (3)

- kA = 2 → (4)

Comparing (1) and (4),

k = -1

Therefore, in (3), from (2),

p = kB

p = -1 × 3

p = -3

Hence the value of p is -3.

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