The sum of the radii of two circles passing through the points $(1,9)$ and $(8,8)$ and are tangent to the x-axis.

Question

There are exactly two circles that pass through the points $(1,9)$ and $(8,8)$ and are tangent to the $x$-axis. Find the sum of their radii.

Is there a general formula for this to find their radii?

Answer 1

Hint: A circle tangent to the $x$-axis has the form $$(x-a)^2+(y-r)^2=r^2.$$

The two points give the two equations$$(1-a)^2+(9-r)^2=r^2\text{ or } 18r=a^2-2a+82$$ $$(8-a)^2+(8-r)^2=r^2 \text{ or } 16r=a^2-16a+128.$$ Eliminate $r$ to get $$-2a^2+256a-992=-2(a-124)(a-4)=0,$$ and substitute back to get $r_1=5,r_2=845,$ making the sum of the radii $850$.