Question

The lines `(1-x)/2 = (y-1)/3 = z/1 and (2x-3)/(2p) = y/-1 = (z-4)/7` are perpendicular to each other for p equal to ______.

Options

• `-1/2`

• `1/2`

• 2

• 3

Answer 1

The lines `(1-x)/2 = (y-1)/3 = z/1 and (2x-3)/(2p) = y/-1 = (z-4)/7` are perpendicular to each other for p equal to 2.

Explanation:

L₁: `(1-x)/2 = (y-1)/3 = z/1`

`(x-1)/-2 = (y-1)/3 = z/1`   ...(i)

L₂: `(2x-3)/(2p) = y/-1 = (z-4)/7`

`(x-3/2)/p = y/-1 = (z-4)/7`    ...(ii)

On Comparing with

`(x-x_1)/a = (y-y_1)/b = (z-z_1)/c`

Direction ratio of line (i) are `a_1 = -2, b_1 = 3, c_1=1`

Direction ratio of line (ii) are `a_2 = p, b_2 = -1, c_2 = 7`

when L₁ ⊥ L₂ then a₁a₂ + b₁b₂ + c₁ c₂ = 0

∴ −2 × p + 3 × (−1) + 1 × 7 = 0

∴ −2p − 3 + 7 = 0

∴ −2p + 4 = 0

∴ p = 2