JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Three Dimensional Geometry

Let

(λ, 2, 1)

be a point on the plane which passes which through the point

(4, - 2, 2)

. If the plane is perpendicular to the line joining the points

(- 2, - 21, 29)

and

(- 1, - 16, 23)

, then

(\frac{λ}{11})^{2} - \frac{4 λ}{11} - 4

is equal to ______

Answer: 8

Solution

The direction ratios of the line joining

(- 2, - 21, 29)

and

(- 1, - 16, 23)

are

(1, 5, - 6)

. This is the normal vector to the plane.

The plane equation is

1 (x - 4) + 5 (y + 2) - 6 (z - 2) = 0

, which simplifies to

x + 5 y - 6 z + 18 = 0

.

Since

(λ, 2, 1)

lies on this plane:

λ + 5 (2) - 6 (1) + 18 = 0

λ + 10 - 6 + 18 = 0

λ = - 22

Now substitute

λ = - 22

into the expression:

(\frac{- 22}{11})^{2} - \frac{4 ( - 22 )}{11} - 4 = (- 2)^{2} + 8 - 4 = 4 + 8 - 4 = 8

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Integer answer

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