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  4. If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 + z2 + 6x - 8y - 2z = 13 and x2 + y2 + z2 -10x + 4y - 2z = 8 then a equals

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Q. If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres $x^2 + y^2 + z^2 + 6x - 8y - 2z = 13$ and $x^2 + y^2 + z^2 -10x + 4y - 2z = 8$ then a equals

Three Dimensional Geometry

Solution:

Plane 2ax - 3ay + 4az + 6 = 0 passes through the mid point of the centre of spheres
$x^2 + y^2 + z^2 + 6x - 8y - 2z = 13$ and
$x^2 + y^2 + z^2 -10x + 4y - 2z = 8$ respectively centre of spheres are (- 3, 4, 1) and (5, - 2, 1). Mid point of centres is (1, 1, 1).
Satisfying this in the equation of plane, we get
2a - 3a + 4a + 6 = 0
$\Rightarrow $ a = -2.