Coordinate Geometry
Chord of contact tangent to ellipse; sum of eccentric angles
MMTS_Full_Test_13
Grade 12
Question:
Chord of contact of tangents from $P,Q$ on $\dfrac{x^2}{9}+\dfrac{y^2}{16}=1$ to $\dfrac{x^2}{3}+\dfrac{y^2}{4}=1$ is normal to $2x^2+2y^2-4x-4y+1=0$. Sum of eccentric angles of $P$ and $Q$ is
(A) $\pi/2$
(B) $\pi$
(C) $2\pi$
(D) none of these
Step-by-Step Solution
Key Concept: Chord of contact of $P(3\cos\alpha,4\sin\alpha)$ to inner ellipse: $x\cos\alpha+y\sin\alpha=1$. Normal to circle $x^2+y^2-2x-2y+1/2=0$ (center $(1,1)$) means chord passes through $(1,1)$: $\cos\alpha+\sin\alpha=1\Rightarrow\alpha=0$ or $\pi/2$.
Sum $=\pi/2$.
Correct Answer: (A) $\pi/2$