Find the locus of the centers of the circles `x^2+y^2-2x-2b y+2=0` , where `a` and `b` are parameters, if the tangents from the origin to each of the circles are orthogonal.

Explanation

The given circle is
`x^(2)+y^(2)-2ax-2by+2=0`
or `(x-a)^(2)+(y-b)^(2)=a^(2)+b^(2)-2`
Its director circle is
`(x-a)^(2)+(y-b)^(2)=2(a^(2)+b^(2)-2)`
Given that tangents drawn from the origin to the circle are orthogonal. It implies that the director circle of the circle must pass through the origin, i.e.,
`a^(2)+b^(2)=2(a^(2)+b^(2)-2)`
or `a^(2)+b^(2)=4`
Thus, the locus of the center of the given circle is `x^(2)+y^(2)=4.`