Suppose the line $\frac{x-2}{\alpha}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3 y-2 z+\beta=0$. Then $(\alpha+\beta)$ is equal to ____.

JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Three Dimensional Geometry

Suppose the line

\frac{x - 2}{α} = \frac{y - 2}{- 5} = \frac{z + 2}{2}

lies on the plane

x + 3 y - 2 z + β = 0

. Then

(α + β)

is equal to ____.

Answer: 7

Solution

The point

(2, 2, - 2)

lies on the line, so it must satisfy the plane:

2 + 3 (2) - 2 (- 2) + β = 0

gives

β = - 12

. The line's direction

(α, - 5, 2)

must be perpendicular to the plane's normal

(1, 3, - 2)

α - 15 - 4 = 0

gives

α = 19

. Therefore

α + β = 19 - 12 = 7

JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Integer answer

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