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

Question

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

Tardigrade Admin · Accepted Answer

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 \to =< k, 2, - 12 >

A normal vector to the first plane

n_{1} \to =< 2, 1, 3 >

A normal vector to the second plane

n_{2} \to =< 1, 2, - 3 >

cos θ = \frac{b \to \cdot n _{i} \to}{∣ ∣ b \to ∣ ∣ ∣ n _{1} \to ∣} = \frac{b \to \cdot n _{2} \to}{∣ ∣ b \to ∣ ∣ ∣ n _{2} \to ∣} \Rightarrow 2 k + 2 - 36 = k + 4 + 36

\Rightarrow k = 74

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

Q. The line $\frac{x}{k} = \frac{y}{2} = \frac{z}{- 12}$ 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