Let P be an arbitrary point on the ellipse fr | vedclass

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Question

(c)

We have, equation of ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Point $P(a \cos 	heta, b \sin 	heta)$ lie on ellipse

Foci of ellips

Vedclass · Accepted Answer

an ellipse

Vedclass · Answer

(c)

We have, equation of ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Point $P(a \cos 	heta, b \sin 	heta)$ lie on ellipse

Foci of ellipse $\left(F_1ight)(a e, 0)$ and $F_2(-a e, 0)$

$	herefore$ Centroid of $	riangle P F_1 F_2=\left(\frac{a \cos 	heta}{3}, \frac{b \sin 	heta}{3}ight)$

$	herefore \quad h=\frac{a \cos 	heta}{3}, K=\frac{b \sin 	heta}{3}$

$\begin{array}{l}\Rightarrow\left(\frac{3 h}{a}ight)^2+\left(\frac{3 k}{b}ight)^2=\cos ^2 	heta+\sin ^2 	heta \\Rightarrow \quad \frac{9 x^2}{a^2}+\frac{9 y^2}{b^2}=1\end{array}$

which represent the locus of ellipse.

Let $P$ be an arbitrary point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $a > b > 0$. Suppose $F_1$ and $F_2$ are the foci of the ellipse. The locus of the centroid of the $\Delta P F_1 F_2$ as $P$ moves on the ellipse is

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