The real numbers a and b are distinct. Consider the circles
`omega_(1): (x-a)^(2)+ (y-b)^(2) = a^(2)+b^(2)` and
`omega_(2): (x-b)^(2) +(y-a)^(2) = a^(2) +b^(2)`
Which of the following is (are) true?

Options

A. A. The line `y = x` is an axis of symmetry for the circles
B. B. The circles intersect at the origin and a point, P(say), which lies on the line `y = x`
C. C. The line `y = x` is the radical axis of the pair of circles.
D. D. The circles are orthogonal for all `a ne b`.

Answer

A, B, C

Explanation

Interchanging x and y in one equation gives the other equation. So, (a) is true,
Clearly (0,0) satisfies both circles and `y = x` is radical axis. So (b) and (c ) are true. The centres are (a,b) and (b,a) the radii are both `sqrt(a^(2)+b^(2))`. The distance between the centres `= |a-b| sqrt(2)`
If circles are orthogonal, `|a-b| sqrt(2) = 2(a^(2)+b^(2))`
Now, `2(a^(2)+b^(2)) ne 2(a^(2)-b^(2))`, for all a and b. Thus, (d) is false.