Question

Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocentre is at the origin, then its third vertex lies in which quadrant?

• A. first
• B. second
• C. third
• D. fourth

Answer 1

The correct option is B second


$m_{AD}\cdot m_{BC}=-1\Rightarrow m_{AH}\cdot m_{BC}=-1$
$\Rightarrow \left(\frac{y-0}{x-0}\right)\cdot \left(\frac{3-2}{4-0}\right)=-1$
$\Rightarrow 4x+y=0\cdots \left(1\right)$
Now,
$m_{BE}\cdot m_{AC}=-1\Rightarrow m_{BH}\cdot m_{AC}=-1$
$\Rightarrow \left(\frac{3-0}{4-0}\right)\cdot \left(\frac{y-2}{x-0}\right)=-1$
$\Rightarrow 4x+3y=6\cdots \left(2\right)$
from the equations $\left(1\right)$ and $\left(2\right),$ we get
$x=-\frac{3}{4}$, $y=3$
Hence, Point $(x,y)$ lies in second quadrant.