Question

The locus of the middle points of chords of an ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$, passing through $P(0,5)$ is another ellipse $E$. The coordinates of foci of the ellipse $E$ is

• A. $(0,\frac{3}{5})$ and $(0,-\frac{3}{5})$
• B. $(0,-2)$ and $(0,1)$
• C. $(0,4)$ and $(0,1)$
• D. $(0,\frac{11}{2})$ and $(0,-\frac{1}{2})$

Answer 1

Answer: (C)

$(0,4)$ and $(0,1)$

$\frac{x^2}{16}+\frac{y^2}{25}=1$

Any point on ellipse is $(4\cos \theta ,5\sin \theta )$

Middle point $(h,k)=(\frac{4\cos \theta +0}{2},\frac{5\sin \theta +5}{2})$

$\cos \theta =\frac{h}{2},\sin \theta =\frac{2k-5}{5}$

The required locus is $\frac{x^2}{4}+\frac{(y-\frac{5}{2}{)}^2}{\frac{25}{4}}=1$

Focii are $(0,\frac{5}{2}+\frac{5}{2}\times \sqrt{1-\frac{16}{25}})$

and $(0,\frac{5}{2}-\frac{5}{2}\times \sqrt{1-\frac{16}{25}})$

$=(0,4)$ and $(0,1)$