Question

A variable line is at constant distance ${}^{\prime }{p}^{\prime }$ from the origin and meets the coordinate axes in ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$. Show that locus of the centroid of $△OAB$ is $x^{-2}+y^{-2}=9P^{-2}.$

Answer 1

$A(\frac{P}{cos\theta },0)$

$B(0,\frac{P}{sin\theta })$

$O(0,0)$

Centroid is given by

$x=\frac{x_1+x_2+x_3}{3}$ $y=\frac{y_1+y_2+y_3}{3}$

$x=\frac{P}{3cos\theta }$

$y=\frac{P}{3sin\theta }$

$cos\theta =\frac{P}{3x}\to \left(1\right)$

$sin\theta =\frac{P}{3y}\to \left(2\right)$

Squaring (1 ) and (2) and then adding

$1=\frac{P^2}{9}(\frac{1}{x^2}+\frac{1}{y^2})$

$\frac{1}{x^2}+\frac{1}{y^2}=\frac{9}{P^2}$