Coordinate Geometry
Focal chord; circumcenter locus
Grade Class 12
Question:
In a parabola $y^2=4ax$, two points $P$ and $Q$ are taken such that the tangents drawn to the parabola at these points meet at the directrix at $R$. The focus (locus of circumcenter of $\triangle PQR$) will be
$(a/2, 0)$
$(a, 0)$
$(3a/2, 0)$
$(5a/2, 0)$
Step-by-Step Solution
Key Concept: When tangents from $P$ and $Q$ meet at the directrix, $PQ$ is a focal chord. Find the circumcenter of $\triangle PQR$ parametrically using $P=(at^2, 2at)$ and the focal chord relation.
PQ is focal chord. Circumcenter locus: $y^2=2a(x-a)$, focus at $(3a/2,0)$.
Correct Answer: 3