Mathematics

The key insight is that the tangent length from a point to a circle depends only on how far that point is from the circle's center, not on where the point sits. Start by rewriting both circles in standard form to identify their centers and radii, then use the tangent-length formula: for a point $P$ outside a circle with center $O$ and radius $r$, the tangent length is $\sqrt{d^2 - r^2}$ where $d$ is the distance from $P$ to $O$. The elegant part is recognizing that any point on the first circle will be equidistant from the second circle's center, making the tangent length constant across all such points.

The key insight is that the tangent length from a point to a circle depends only on how far that point is from the circle's center, not on where the point sits. Start by rewriting both circles in standard form to identify their centers and radii, then use the tangent-length formula: for a point $P$ outside a circle with center $O$ and radius $r$, the tangent length is $\sqrt{d^2 - r^2}$ where $d$ is the distance from $P$ to $O$. The elegant part is recognizing that any point on the first circle will be equidistant from the second circle's center, making the tangent length constant across all such points.