JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Three Dimensional Geometry

If the equation of the plane passing through the line of intersection of the planes

2 x - 7 y + 4 z - 3 = 0, 3 x - 5 y + 4 z + 11 = 0

and the point

(- 2, 1, 3)

a x + b y + cz - 7 = 0

, then the value of

2 a + b + c - 7

is ____

Answer: 4

Solution

Every plane through the line of intersection has the form

(2 x - 7 y + 4 z - 3) + λ (3 x - 5 y + 4 z + 11) = 0.

(- 2, 1, 3)

the first bracket is

- 4 - 7 + 12 - 3 = - 2

and the second is

- 6 - 5 + 12 + 11 = 12

, so

- 2 + 12 λ = 0

and

λ = \frac{1}{6}

. Multiplying the resulting equation by

6

clears the fractions and makes the constant term

- 7

15 x - 47 y + 28 z - 7 = 0.

a = 15

b = - 47

c = 28

, and

2 a + b + c - 7 = 30 - 47 + 28 - 7 = 4

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Integer answer

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