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  •    is a variable focal chord of the parabola  <img alt="\mathrm{ y^2=4 a x }

\mathrm{P Q}   is a variable focal chord of the parabola  \mathrm{ y^2=4 a x }  whose vertex is  \mathrm{ A }  . Then the locus of the centroid of triangle \mathrm{ APQ } 

Option: 1

is a parabola whose latus rectum is \mathrm {\frac{9}{12} a.}
 


Option: 2

is a parabola whose latus rectum is  \mathrm {\frac{12}{9} a.}
 


Option: 3

is a parabola whose latus rectum is  \frac{3 \mathrm a}{9}.
 


Option: 4

None of these


Answers (1)

PQ being focal chord,  \mathrm {t_1 t_2=-1 \: \: or \: \: t_2=-1 / t_1.}  If  \mathrm {(x, y)} be the centroid

of  \mathrm {\triangle A P Q,} then

 \mathrm { 3 x=a\left(t_1^2+\frac{1}{t_1^2}\right), 3 y=2 a\left(t_1-\frac{1}{t_1}\right) }

Eliminating  \mathrm { t_1}, we get

\begin{aligned} & \mathrm {9 y^2=4 a(3 x-2 a)} \\ & \mathrm {y^2=\frac{4 a \cdot 3}{9}\left(x-\frac{2 a}{3}\right)} \end{aligned}
or
It is a parabola whose L.R. is   \mathrm {\frac{12}{9} a=\frac{4}{3} a}

Posted by

Ramraj Saini

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