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 an - Mathematics MCQ AIEEE 2026 | ExamTest.live

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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:

Solution & Step-by-step Explanation

Centres of the spheres:Sphere 1: x^{2} + y^{2} + z^{2} + 6 x - 8 y - 2 z = 13 ⟹ C_{1} = (- 3, 4, 1) .Sphere 2: x^{2} + y^{2} + z^{2} - 10 x + 4 y - 2 z = 8 ⟹ C_{2} = (5, - 2, 1) .Midpoint of C_{1} C_{2} : M = (\frac{- 3 + 5}{2}, \frac{4 - 2}{2}, \frac{1 + 1}{2}) = (1, 1, 1) The plane 2 a x - 3 a y + 4 a z + 6 = 0 passes through (1, 1, 1) : 2 a (1) - 3 a (1) + 4 a (1) + 6 = 0 2 a - 3 a + 4 a + 6 = 0 3 a + 6 = 0 ⟹ 3 a = - 6 ⟹ a = - 2

Try it yourself before checking the explanation above.

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:

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1

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2

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