Question

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

• A. $(4,7)$
• B. $(-4,-7)$
• C. $(2,-3)$
• D. $(5,-1)$

Answer 1

The correct option is B $(-4,-7)$

Let the given points be
$A=(5,-1),B=(-2,3),H=(0,0)$


We know that
$m_{CH}\times m_{AB}=-1\Rightarrow \frac{y-0}{x-0}\times \frac{-4}{7}=-1\Rightarrow y=\frac{7x}{4}\cdots \left(1\right)$
Also,
$m_{BC}\times m_{AH}=-1\Rightarrow \frac{y-3}{x+2}\times \frac{-1}{5}=-1\Rightarrow y-3=5x+10$
Using equation $\left(1\right)$, we get
$\Rightarrow \frac{7x}{4}=5x+13\Rightarrow x=-4\Rightarrow y=-7$

Hence, the coordinates of the vertex are $(-4,-7).$