Question

Let $P$ be any point on ellipse $3x^2+4y^2=12$ and $S,S$ are its foci then the locus of the centroid of triangle PSS' is a conic $C$ whose

• A. length of latus rectum equals 1 .
• B. locus of point of intersection of perpendicular tangents is $x^2+y^2=\frac{7}{9}$
• C. equation of auxiliary circle is $x^2+y^2=\frac{4}{9}$
• D. area of quadrilateral formed by tangents at the extremities of latus-rectum equals $\frac{16}{3}$.

Answer 1

Answer: (A), (B), (C)

$\frac{x^2}{4}+\frac{y^2}{3}=1;e^2=1-\frac{3}{4}=\frac{1}{4}\Rightarrow e=\frac{1}{2}$

$\therefore 3h=2\cos \theta \Rightarrow \cos \theta =\frac{3h}{2}$
and $3k=\sqrt{3}\sin \theta \Rightarrow \sin \theta =\sqrt{3}k$
Now, on squaring and adding, we get
$1=\frac{9x^2}{4}+3y^2\text{ or }\frac{x^2}{\frac{4}{9}}+\frac{y^2}{\frac{1}{3}}=1$
Now verify alternatives.