The plane px + y + pz = r intersects the co-ordinate axes at A, B and C. Find the co-ordinates of…

$A\left(\frac{r}{p}, 0, 0\right)$, $B(0, r, 0)$, $C\left(0, 0, \frac{r}{p}\right)$

Let the orthocentre be $O(a, b, c)$. $OA \perp BC \Rightarrow c = bp$ $OB \perp AC \Rightarrow a = c$ $\therefore a = bp = c \Rightarrow a = \frac{b}{1/p} = c = k$ (say) Since, the orthocentre lies on the plane $px + y + pz = r$, $pk + \frac{k}{p} + pk = r \Rightarrow k = \frac{r}{2p + \frac{1}{p}} = \frac{rp}{2p^2+1}$ Co-ordinates of the orthocentre is $\left(\frac{rp}{2p^2+1}, \frac{r}{2p^2+1}, \frac{rp}{2p^2+1}\right)$.