Ellipse
Circle Inscribed in Chords of Family of Ellipses
nta_pyq_2023_apr
Grade 11
Question:
Consider ellipses $E_k:\ kx^2+k^2y^2=1$, $k=1,2,\ldots,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and one on major axis) of $E_k$. If $r_k$ is the radius of $C_k$, then $\displaystyle\sum_{k=1}^{20}\dfrac{1}{r_k^2}$ is equal to
Step-by-Step Solution
Key Concept: $E_k:\ \frac{x^2}{1/k}+\frac{y^2}{1/k^2}=1$. End points of axes: $(\frac{1}{\sqrt{k}},0)$ and $(0,\frac{1}{k})$. The chord has equation $\sqrt{k}x+ky=1$. Distance from origin $=r_k=\frac{1}{\sqrt{k+k^2}}$.
$\sum\frac{1}{r_k^2}=\sum(k+k^2)=210+2870=3080$.
Correct Answer: 1