Mathematics

Mathematics

Problem

If the plane 2ax−3ay+4az+6=02ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2+y2+z2+6x−8y−2z=13x^2 + y^2 + z^2 + 6x - 8y - 2z = 13 and x2+y2+z2−10x+4y−2z=8x^2 + y^2 + z^2 - 10x + 4y - 2z = 8, then aa equals
Intuitive explanation

The key is to extract the sphere centres from their general equations, then find the midpoint of the segment joining them—this midpoint must satisfy the plane equation. Rewrite each sphere in standard form by completing the square to read off the centre coordinates directly, then substitute the midpoint into the plane equation to solve for aa.

Intuitive explanation

The key is to extract the sphere centres from their general equations, then find the midpoint of the segment joining them—this midpoint must satisfy the plane equation. Rewrite each sphere in standard form by completing the square to read off the centre coordinates directly, then substitute the midpoint into the plane equation to solve for aa.

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