question_answer

Let P be arbitrary point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-1=0,a>b>0.\]Suppose \[{{F}_{1}}\] and \[{{F}_{2}}\] are the foci of the ellipse. The locus of the centroid of the triangle \[P{{F}_{1}}{{F}_{2}}\] as P moves on the 1 ellipse, is

A) a circle           

B)    a parabola

C) an ellipse       

D) a hyperbola

Correct Answer: C

Solution :

\[\therefore \,\,h=\frac{a\,\,\cos \,\theta }{3}\Rightarrow \,\cos \,\theta =\frac{3h}{a}\]                 \[k=\frac{b\,\sin \,\theta }{3}\Rightarrow \,\sin \,\theta \,=\frac{3k}{b}\]                 \[\therefore \,\,\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{1}{9}\](ellipse)