Ellipse
Circle-Ellipse Parametric — Eccentricity
nta_pyq_2026_jan
Grade 11
Question:
Let $(h,k)$ lie on the circle $C:x^2+y^2=4$ and the point $(2h+1,3k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\dfrac{5}{e^2}$ is equal to _____.
Step-by-Step Solution
Key Concept: $(h,k)$ on $x^2+y^2=4$: $h=2\cos\theta$, $k=2\sin\theta$. Transformed point: $(4\cos\theta+1,6\sin\theta+2)$. Let $X=4\cos\theta+1$, $Y=6\sin\theta+2$: $\dfrac{(X-1)^2}{16}+\dfrac{(Y-2)^2}{36}=1$.
$e^2=\tfrac{5}{9}$. $\dfrac{5}{e^2}=9$.
Correct Answer: 9