A variable chord is drawn through the origin to the circle `x^2+y^2-2a x=0` . Find the locus of the center of the circle drawn on this chord as diameter.

Options

A. `x^(2)+y^(2)-ax=0`
B. `x^(2)+y^(2)+ax=0`
C. `x^(2)+y^(2)-ay=0`
D. `x^(2)+y^(2)-ax-ay=0`

Answer

A

Explanation

Let variable chord be `y=mx`
Equation of a circle passes through the intersection of a circle and a line
`x^(2)+y^(2)-2ax+lamda(mx-y)=0`
`x^(2)+y^(2)-(2a-lamdam)x-lamday=0`
Centre of this circle `((2a-lamdam)/2,(lamda)/2)`
Lie on the line `y=mx`
`:." "(lamda)/2=m((2a-lamdam)/2)`......(i)
Let centre be `(h,k)`
`:.h=(2a-lamdam)/2`........(ii)
`k=(lamda)/2`......(iii)
Eliminate `lamda` and m from equation (i), (ii) and (iii)
`h^(2)+k^(2)-ah=0`
`:." "` Locus of (h,k)
`x^(2)+y^(2)-ax=0`