Question

If the vertices of a triangle are $(\sqrt{5},0),(\sqrt{3},\sqrt{2})$ and $(2,1)$, then the orthocentre of the triangle is

• A. $(\sqrt{5},0)$
• B. $(0,0)$
• C. $(\sqrt{5}+\sqrt{3}+2,\sqrt{2}+1)$
• D. None of these

Answer 1

Answer: (C)

$(\sqrt{5}+\sqrt{3}+2,\sqrt{2}+1)$

The circumcentre $O$ of the triangle is $(0,0)$ as all the vertices lie on the circle $x^2+y^2=5$
Centroid $G$ divides $OH$ in the ratio $1:2$
$H$ is the orthocentre
Let $H$ be $(x,y)$ and $G$ be $(x_0,y_0)$
then $x_0=\frac{\sqrt{5}+\sqrt{3}+2}{3}=\frac{x}{3},y_0=\frac{\sqrt{2}+1}{3}=\frac{y}{3}$
$x=\sqrt{5}+\sqrt{3}+2,y=\sqrt{2}+1$
Hence, the correct option is $C$.