Ellipse
Chord with Given Midpoint
nta_pyq_2024_jan
Grade 11
Question:
The length of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose mid point is $\left(1,\dfrac{2}{5}\right)$, is equal to:
$\dfrac{\sqrt{1691}}{5}$
$\dfrac{\sqrt{2009}}{5}$
$\dfrac{\sqrt{1741}}{5}$
$\dfrac{\sqrt{1541}}{5}$
Step-by-Step Solution
Key Concept: Use the equation of chord with midpoint $(x_1,y_1)$: $T=S_1$. Find the chord equation, substitute back into the ellipse to get a quadratic, then compute the distance between the two endpoints.
Chord eqn: $\frac{x}{25}+\frac{2y/5}{16\cdot2}=\frac{1}{25}+\frac{4/25}{16}$... Using $T=S_1$: $8x+5y=10$. Substituting gives $4x^2-8x-15=0$. Endpoints give chord length $=\frac{\sqrt{1691}}{5}$.
Correct Answer: 1