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

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) (1/11) (B) (5/11) (C) 6 (D) 7

Tardigrade Admin · Accepted Answer

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 + λ (3 x - y + z - 5) = 0

\frac{x}{\frac{5 λ + 1}{2 + 3 λ}} + \frac{y}{\frac{5 λ + 1}{1 - λ}} + \frac{z}{\frac{5 λ + 1}{λ - 5}} = 1

Given

\frac{5 λ + 1}{2 + 3 λ} = 1 \Rightarrow λ = \frac{1}{2}

So intercept on

y -

axis

= \frac{5 λ + 1}{1 - λ} = 7

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

$\frac{1}{11}$

$\frac{5}{11}$

6

7

Q. If the plane P passes through the intersection of two mutually perpendicular planes 2x+ky −5z=1 and 3kx−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

111​

115​

6

7

Two given planes mutually perpendicular 2(3k)+k(−k)+(−5)1=0 k=1,5 but k<3 So k=1 Plane passing through these planes is 2x+y−5z−1+λ(3x−y+z−5)=0 2+3λ5λ+1​x​+1−λ5λ+1​y​+λ−55λ+1​z​=1 Given 2+3λ5λ+1​=1⇒λ=21​ So intercept on y− axis =1−λ5λ+1​=7

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

$\frac{1}{11}$

$\frac{5}{11}$