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. Fourth
• B. Second
• C. Third
• D. First

Answer 1

The correct option is B Second

$m_{BD}\times m_{AD}=-1\Rightarrow \left(\frac{3-2}{4-0}\right)\times \left(\frac{b-0}{a-0}\right)=-1$

$\Rightarrow b+4a=0$ ...(i)

$m_{AB}\times m_{CF}=-1\left(\frac{(b-2)}{a-0}\right)\times \left(\frac{3}{4}\right)=-1$

$\Rightarrow 3b-6=-4a\Rightarrow 4a+3b=6$ ...(ii)

From (i) and (ii)

$a=\frac{-3}{4},b=3$

$\therefore I{I}^{nd}$ quadrant.