Coordinate Geometry
Circle tangent to line; intersects given circle; diameter condition
MMTS_Full_Test_10
Grade 12
Question:
Circle $C$ touches $x=2y$ at $(2,1)$ and intersects $C_1:x^2+y^2+2y-5=0$ at two points $P,Q$ such that $PQ$ is a diameter of $C_1$. Length of diameter of $C$ is
(A) $7\sqrt5$
(B) 15
(C) $\sqrt{285}$
(D) $4\sqrt{15}$
Step-by-Step Solution
Key Concept: $C$ tangent to $x=2y$ at $(2,1)$: center lies on normal to $x-2y=0$ through $(2,1)$, i.e., line $2x+y=5$. Also $PQ$ diameter of $C_1$ means $PQ$ passes through center of $C_1$. Center of $C_1=(0,-1)$. Radical axis of $C$ and $C_1$ is $PQ$. Find $C$, then diameter.
Diameter of $C=7\sqrt5$.
Correct Answer: (A) $7\sqrt5$