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Question

P(a,b) is a point in the first quadrant. Circles are drawn through P touching the coordinate axes such that the length of common chord of these circles is maximum, if possible values of a/b is k1 and k2, then k1+k2 is equal to

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Solution

Let r1,r2 be the radii of the circles.
Equations will be of the form x2+y22r(x+y)+r2=0
it passes through P(a,b)
a2+b22r(a+b)+r2=0r1,r2=a+b+2ab,a+b2ab
Now the common chord equation is
S1S2=0x+y=a+b
For maximum length of common chord , common chord becomes diameter of smaller circle i.e., (a+b2ab,(a+b2ab) should lie on common chord.
putting the coordinates in chord equation
We get, a+b=22aba2+b2=6ab(ab)2+ab6=0
So from sum of the roots
k1+k2=6

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