Circle equations

Question

Given that the circle C has center $(a,b)$ where $a$ and $b$ are positive constants and that C touches the $x$-axis and that the line $y=x$ is a tangent to C show that $a = (1 + \sqrt{2})b$

Answer 1

Easy to solve using simple geometry, write sin (45º)

Answer 2

Your circle has to be tangent both to the $x$-axis and to the line $y=x$. These two straight lines form 4 angles, and the center of the circle lies on the bisectrix of the (only) angle which is contained in the first quadrant. This is because the distance of the center from both lines is the same (the distance from the center of a circle to a line tangent to the circle is the radius). From here it should be downhill...

Without using trigonometry, you can also consider the point $A$ of tangency between $y=x$ and the circle. Take the line $r$ which passes through A and the center of the circle, and call $B$ the intersection of this line with the $x$-axis. Call $O$ the origin; what can you say about the triangle $OAB$?