IIT-JEE 2009 · Paper 2, Mathematics question 23

Vertices. The lines meet $y=0$ at $A(-p,0)$ and $B(-q,0)$. Written as $y=\frac{1+p}p(x+p)$ and $y=\frac{1+q}q(x+q)$, they meet where
$$(1+p)q(x+p)=(1+q)p(x+q)\implies x(q-p)=pq(q-p)\implies x=pq.$$

Orthocentre. The altitude from the third vertex is vertical, $x=pq$. The altitude from A is perpendicular to the second line: $y=-\frac q{1+q}(x+p)$, which at $x=pq$ gives $y=-pq$.

Locus. $H=(pq,-pq)$ lies on $x+y=0$: a straight line.

(D)