Let C_1 be the circle in the third quadrant of radius 3 that touches both coordinate axes.

Question

Let $C_1$ be the circle in the third quadrant of radius $3$ that touches both coordinate axes. Let $C_2$ be the circle with centre $(\alpha,\beta)$ that touches $C_1$ externally. If $(\beta-\alpha)^2=\dfrac{m}{n}$ (with $\gcd(m,n)=1$), then $m+n$ is equal to:

Accepted Answer

*(The precise auxiliary conditions on $C_2$ are corrupted in the source; $C_1$ has centre $(-3,-3)$. The recoverable working converges on $m+n=22$, matching option A.)*

*(Note: the source PDF's question numbering runs continuously; Questions 112 onward, appearing next in sequence, are area-under-curve / differential-equation items that appear to have been merged into this "Coordinate Geometry" bank in the source PDF itself. They are reproduced below exactly as numbered, for completeness.)*

Solutions · 1