Question

In a $△OAB$, where $O$ is origin and vertex $B$ is $(3,4)$. If the orthocentre of the triangle is $P(1,4)$ and coordinates of vertex $A$ is $(h,k)$, then the value of $4k+h$ is

Answer 1

As $P$ is the orthocentre of $△OAB$,
$m_{OA}\cdot m_{BP}=-1$
$\because m_{BP}=0\Rightarrow m_{OA}=\frac{k}{h}$is undefined $\Rightarrow h=0$
Also,
$m_{OB}\cdot m_{AP}=-1\Rightarrow \frac{4}{3}\times \frac{4-k}{1-h}=-1\Rightarrow 16-4k=3h-3\Rightarrow k=\frac{19}{4}$

So, the point $A$ is
$(h,k)=(0,\frac{19}{4})$
$\therefore 4k+h=19$