Coordinate Geometry
Common chord of two circles
Grade Class 12
Question:
Circles $C_1:x^2+y^2=625$, $C_2:(x-a)^2+y^2=576$, $a\in(1,49)$. Point $P$ is on both circles with $\angle QPR=\cos^{-1}\!\left(\dfrac{\sqrt{481}}{25}\right)$ ($Q,R$ are centers). Length of common tangent $=\sqrt{1295}$. Common chord length is
Step-by-Step Solution
Key Concept: From common tangent length: $\sqrt{a^2-(r_1-r_2)^2}=\sqrt{1295}\Rightarrow a^2-1=1295\Rightarrow a=36$. Radical axis $x=(a^2-49)/(2a)$. Chord $=2\sqrt{r_1^2-h^2}$.
Common chord $=16$.
Correct Answer: 2