Circles
Circumradius of triangle formed by common point and tangent endpoints
MJAT_TS4_P1
Grade 12
Question:
A line touches the circles $x^2+y^2=81$ and $x^2+y^2-16x-16y+112=0$ at $A$ and $B$ respectively. $P$ is one of the points common to both circles and is farthest from $AB$. The circumradius of $\triangle PAB$ is:
Step-by-Step Solution
Key Concept: Find the common external/internal tangent. The circumradius of $\triangle PAB$: use the formula $R=\frac{PA\cdot PB\cdot AB}{4\cdot\text{Area}}$. Since $PA$ and $PB$ are related to the tangent lengths from $P$ to each circle.
Circumradius $=\mathbf{6}$.
Correct Answer: 6