A Plane Passes Through A Fixed Point (a, B, C) And Cuts Axes In A, B, C. Find The Locus Of The Centre Of The Sphere OABC.

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Question

A plane passes through a fixed point (a, b, c) and cuts axes in A, B, C. Find the locus of the centre of the sphere OABC.

diffcult

Solution

Correct option is

Let the equation of a plane passing through the fixed point (a, b, c) be

…(1)

…(2)

Putting …(2)

Putting y = 0, z = 0, in (1), we get px = 1 i.e., x = 1/p.

∴ A = (1/p, 0, 0). Similarly B = (0,1/q, 0), C=(0, 0, 1/r).

Let the equation of the sphere passing through O, A, B, C, be

…(3)

(0, 0, 0) is on the sphere (3).

A is on the sphere (3).

Similarly,

Let the centre of this sphere be ().

Putting the values of p, q, r in (2), we get