P is a point a , b in the first quadrant. If the two circles which pass through P and touch both the coordinate axes, cut at right angles, then :

Step-by-step solution

Let the equation of circle which touches both coordinate axes is x - r 2 + y - r 2 = r 2 ⇒ x 2 + y 2 - 2 r x - 2 r y + r 2 = 0 where, r is radius. It is passing through a , b , so r 2 - 2 ( a + b ) r + a 2 + b 2 = 0 Let the roots of this equation are r 1 , r 2 Sum of roots r 1 + r 2 = 2 a + b Product of roots r 1 · r 2 = a 2 + b 2 Both circles intersect orthogonally each other so 2 r 1 r 2 + 2 r 1 r 2 = r 1 2 + r 2 2 ⇒ 6 r 1 r 2 = r 1 + r 2 2 ⇒ 6 a 2 + b 2 = 4 ( a + b ) 2 ⇒ a 2 - 4 a b