JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Three Dimensional Geometry

The acute angle between the planes

P_{1}

and

P_{2}

, when

P_{1}

and

P_{2}

are the planes passing through the intersection of the planes

5 x + 8 y + 13 z - 29 = 0

and

8 x - 7 y + z - 20 = 0

and the points

(2, 1, 3)

and (

0, 1, 2

), respectively, is

Answer: (A)

Solution

The plane through the intersection of

5 x + 8 y + 13 z - 29 = 0

and

8 x - 7 y + z - 20 = 0

5 x + 8 y + 13 z - 29 + λ (8 x - 7 y + z - 20) = 0

.

For

P_{1}

passing through

(2, 1, 3)

: substituting gives

λ = \frac{7}{2}

, yielding

P_{1} : 2 x - y + z = 6

.

For

P_{2}

passing through

(0, 1, 2)

: similarly, we get

P_{2} : x + y + 2 z = 5

.

The angle between the planes is

θ = cos^{- 1} (\frac{∣2 \cdot 1 + ( - 1 ) \cdot 1 + 1 \cdot 2∣}{6 6}) = cos^{- 1} (\frac{3}{6}) = cos^{- 1} (\frac{1}{2}) = \frac{π}{3}

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option