Question

The equation of the circle passing through the point(1,1) &having two diameters along pair of lines x^2-y^2-2x+4y-3=0 is

Answer 1

The diameters must be intersecting at the center of the circle.Hence, you should first find the center of the circle.
You also know the circle passes through (1,1) , hence with center known, you can find the radius of the circle.
With both center and radius known, you should be able to find the equation of the circle easily.