The line x k = y 2 = z - 12 form an isosceles triangle with the planes 2 x + y + 3 z - 1 = 0 and x + 2 y - 3 z - 1 = 0 , then the value of k is equal to

Step-by-step solution

If given line makes an isosceles triangle with the planes, then the angle that the line makes with the normal vectors to the planes are equal A vector parallel to the line is b → = < k , 2 , - 12 > A normal vector to the first plane n 1 → = < 2 , 1,3 > A normal vector to the second plane n 2 → = < 1 , 2 , - 3 > cos θ = b → ⋅ n i → b → n 1 → = b → ⋅ n 2 → b → n 2 → ⇒ 2 k + 2 − 36 = k + 4 + 36 ⇒ k = 74