JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Three Dimensional Geometry

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

Answer: (D)

Solution

For mutually perpendicular planes, the dot product of their normals is zero:

2 (3 k) + k (- k) + (- 5) (1) = 0

6 k - k^{2} - 5 = 0

k^{2} - 6 k + 5 = 0

, giving

k = 1

k = 5

.

Since

k < 3

, we have

k = 1

.

The plane through the intersection is:

2 x + y - 5 z - 1 + λ (3 x - y + z - 5) = 0

For unit intercept on

x

-axis, setting

y = z = 0

\frac{5 λ + 1}{2 + 3 λ} = 1

, giving

λ = \frac{1}{2}

.

The

y

-intercept is:

\frac{5 λ + 1}{1 - λ} = \frac{5 ( \frac{1}{2} ) + 1}{1 - \frac{1}{2}} = \frac{3.5}{0.5} = 7

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option