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

Answer: 2.5

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