Question

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$, and if its orthocentre lies at the origin, the coordinates of the third vertex are?

Answer 1

Slope of $AO=\frac{-1-0}{5-0}=-\frac{1}{5}$

Slope of line BC = $5$

$y=5x+c$

$3=-10+c$

$⟹c=13$

$y=5x+13$

Slope of $BO=\frac{3-0}{-2-0}=-\frac{3}{2}$

Slope of line $AC=\frac{2}{3}$

$3y=2x+c$

$-3=10+c$

$c=-13$

$3y=2x-13$

$15x+39=2x-13$

$13x=-52$

$x=-4$

$y=-7$