If a circle Passes through a point (1,2) and cut the circle `x^2+y^2 = 4` orthogonally,Then the locus of its centre is

Explanation

Given circle is `x^(2)+y^(2)=4`.
Centre of the circle is `C_(1)(0,0)` and radius `r_(1)=2`.
Let the centre of the variable circle touching the given circle be `C_(2)(h,k)` and radius be `r_(2)`.
We have
`C_(1)C_(2)=r_(1)-r_(2)` (1)
Since variable circle passes through point P(1,0), its radius `r_(2)=C_(2)P=sqrt((h-1)^(2)+k^(2))`
From (1), we have
`sqrt(h^(2)+h^(2))=2-sqrt((h-1)^(2)+k^(2))` `( :' `point (1,0) lies inside the given circle )
`implies sqrt((h-1)^(2)+k^(2))=2-sqrt(h^(2)+k^(2))`
`implies h^(2)+k^(2)-2h+1=4+h^(2)+k^(2)-4sqrt(h^(2)+k^(2))` ,brgt `implies 3+2h=4sqrt(h^(2)+k^(2))`
`implies 9=4h^(2)+12h=16h^(2)+16k^(2)`
`implies 12x^(2)+16y^(2)-12x=9`
This is the required locus.