Question

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.`

Answer 1

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)`.
So, equation through plane at point (a,b,c) will be,
`a/(2alpha)+b/(2beta)+c/(2gamma) = 1`
`=> a/alpha+b/beta+c/gamma = 2`
If, we replace `(alpha,beta,gamma)` by `(x,y,z)`, equation will be:
`a/x+b/y+c/z = 2`