Circles are drawn through $P$ touching the coordinate axes,such that the length of the common chord of these circles is maximum.Find the ratio $a:b$

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 the common chord of these circles is maximum.Find the ratio $a:b$.

The equation of the circle which touches both the coordinate axes is $x^2+y^2-2xr-2yr+r^2=0$.And as this circle passes through $P(a,b)$.

So $a^2+b^2-2ar-2br+r^2=0$
I am stuck here.I do not know what is the equation of length of the common chord of these circles.Please help me.

Answer 1

For each $P$ there will be two circles drawn through it that will touch both the axes and their centres will obviously lie on the line $y=x$ as shown:

As is evident from the figure, the common chord $PQ$ will always be the line segment joining the point $P$ to the point $Q$ which is the reflection of $P$ with respect to the line $y=x$ . Hence the coordinate of $Q\equiv(b,a)$ .

Now to maximise the length of the common chord, the chord $PQ$ should be a diameter of the circle passing through $P$ with it's centre at $\left(\dfrac{a+b}{2},\dfrac{a+b}{2}\right)$ and this circle must also touch both the axes. Can you find the answer now?

EDIT: The equation of the circle with PQ as its diamter is : $$(x-a)(x-b)+(y-a)(y-b)=0$$

If on solving this circle with the say, the X axis, if there are repeated roots, then it will touch said axis. so we get the condition: $$(a+b)^2=8ab$$