A variable plane passes through a fixed point `(a ,b ,c)` and cuts the coordinate axes at points `A ,B ,a n dCdot` Show that eh locus of the centre of the sphere `O A B Ci s a/x+b/y+c/z=2.`

Explanation

Let `alpha,beta,gamma` are coordinates of the locus at any point. So, at the centre of the sphere, equation can be given as:

`(x-alpha)^2+(y-beta)^2+(c-gamma)^2 = alpha^2+beta^2+gamma^2`
(Here, `alpha^2+beta^2+gamma^2 = (radius)^2` as circle is passing through origin)

Solving the above equation, we get,

`=>x^2+y^2+z^2-2alphax-2betay-2gammaz=0`

At x-axis, it will be, `x^2 = 2alphax` as y and z will be 0.

`x=2alpha`

So, point at x-axis will be `(2alpha,0,0)`.

Similarly, at y-axis points will be`(0,2beta,0)` and at z-axis, point will be `(0,0,2gamma)`.

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