Question

Let $P$ be any point on ellipse $3 x^2+4 y^2=12$ 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)

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

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