JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Three Dimensional Geometry

Let

P_{1} : r \cdot (2 \hat{i} + \hat{j} - 3 \hat{k}) = 4

be a plane. Let

P_{2}

be another plane which passes through the points ( 2 , -3,2)(2,-2,-3) and (1,-4,2). If the direction ratios of the line of intersection of

P_{1}

and

P_{2}

be 16,

α, β

, then the value of

α + β

is equal to ____.

Answer: 28

Solution

Plane

P_{1}

2 x + y - 3 z = 4

.

For plane

P_{2}

through

(2, - 3, 2)

(2, - 2, - 3)

, and

(1, - 4, 2)

:

Using the determinant form:

x - 2 0 - 1 y + 3 1 - 1 z - 2 - 5 0 = 0

Expanding:

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

- 5 x + 5 y + z + 23 = 0

The direction ratios of the line of intersection are proportional to:

n_{1} \times n_{2} = (2, 1, - 3) \times (- 5, 5, 1) = (16, 13, 15)

α = 13

and

β = 15

.

Therefore:

α + β = 28

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Integer answer

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