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 ______.

विकल्प

• `-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:

`L_1: (1-x)/2 = (y-1)/3 = z/1`

`(x-1)/-2 = (y-1)/3 = z/1`   ...(i)

`L_2: (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-4)/7`

Direction ratio of line (i) are

a₁ = −2, b₁ = 3, c₁ = 1

Direction ratio of line (ii) are

a₂ = p, b₂ = −1, c₂ = 7

when L₁ ⊥ L₂ then a₁a₂ + b₁b₂ + c₁c₂ = 0

∴ −2 × p + 3 × (−1) + 1 × 7 = 0

∴ −2p − 3 + 7 = 0

∴ −2p + 4 = 0

∴ p = 2