Hi Mohammad,

The key observation here is that the center of your circle must be on the line $y=x$ (that goes through the origin and through the midpoint of $(a,b)$ and $(b,a)$). A typical point on this line is $(c, c),$ so your first job is to find the equation of the circle with center $(c, c)$ that passes through the origin — you can almost do that in your head! (Did you get $x^2 + y^2 -2c(x+y) = 0 ?$)

To find out what $c$ is in terms of $a$ and $b,$ simply replace $x$ by $a$ and $y$ by $b$ in your equation of the circle, then solve for $c.$ You end up with the equation of a circle with $c$ replaced by an expression involving $a$ and $b.$

To find the chord lengths plug $y=0$ into the equation of the circle and solve for $x.$

Chris