JEE Main · 2005 · 08 May 2005 · Offline · Three Dimensional Geometry

If the plane

2 a x - 3 a y + 4 a z + 6 = 0

passes through the midpoint of the line joining the centres of the spheres

x^{2} + y^{2} + z^{2} + 6 x - 8 y - 2 z = 13

and

x^{2} + y^{2} + z^{2} - 10 x + 4 y - 2 z = 8

, then

a

equals

(A)-1
(B)1
(C)-2✓ Correct
(D)2

Answer: (C)

Solution

Completing squares, the sphere

x^{2} + y^{2} + z^{2} + 6 x - 8 y - 2 z = 13

has centre

(- 3, 4, 1)

, and the sphere

x^{2} + y^{2} + z^{2} - 10 x + 4 y - 2 z = 8

has centre

(5, - 2, 1)

. Their midpoint is

(1, 1, 1)

. Requiring the plane

2 a x - 3 a y + 4 a z + 6 = 0

to pass through this point gives

2 a - 3 a + 4 a + 6 = 0

, that is

3 a + 6 = 0

, so

a = - 2

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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