Question

If the plane P passes through the intersection of two mutually perpendicular planes $2 x + ky$ $-5 z =1$ and $3 kx - ky + z =5, k <3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is

• A. $\frac{1}{11}$
• B. $\frac{5}{11}$
• C. 6
• D. 7

Answer 1

Answer: (D)

Two given planes mutually perpendicular
$ 2(3 k )+ k (- k )+(-5) 1=0$
$k =1,5$
but $k <3 $ So $k =1$
Plane passing through these planes is
$2 x + y -5 z -1+\lambda(3 x - y + z -5)=0$
$\frac{x}{\frac{5 \lambda+1}{2+3 \lambda}}+\frac{y}{\frac{5 \lambda+1}{1-\lambda}}+\frac{z}{\frac{5 \lambda+1}{\lambda-5}}=1$
Given $\frac{5 \lambda+1}{2+3 \lambda}=1 \Rightarrow \lambda=\frac{1}{2}$
So intercept on $y-$ axis $=\frac{5 \lambda+1}{1-\lambda}=7$