Question

The distance between the circumcentre and orthocentre of the triangle whose vertices are $(0,0),(6,8)$ and $(-4,3)$ is $L$ then the value of $\frac{L}{\sqrt{5}}$ is_______.

Answer 1

Given vertices of triangle are $O(0,0), B(6,8)$ and $C(-4,3)$.
Slope of $O B=\frac{8}{6}=\frac{4}{3}$
Slope of $O C=-\frac{3}{4}$
$\therefore \angle B O C=\frac{\pi}{2}$
$\triangle O B C$ is right angled at $O$.
Circumcentre $=$ mid point of hypotenuse $B C=\left(1, \frac{11}{2}\right)$
Orthocentre $=$ vertex $O(0,0)$
Required distance $=\sqrt{\left(1+\frac{121}{4}\right)}=\frac{5 \sqrt{5}}{2}$ unit