Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
The straight line joining any point $P$ on the parabola $y^2 = 4ax$ to the vertex and perpendicular from the focus to the tangent at $P$, intersect at $R$, then the equation of the locus of $R$ is:
$x^2 + 2y^2 - ax = 0$
$2x^2 + y^2 - 2ax = 0$
$2x^2 + 2y^2 - ay = 0$
$2x^2 + y^2 - 2ay = 0$
Step-by-Step Solution
Key Concept: The locus is found by eliminating the parameter from the tangent line and the perpendicular from a fixed point.
The tangent at point $P$ on the parabola is $y = x + at^2$. The line perpendicular to this tangent passing through $(a, 0)$ is $y = -t(a - x)$ or $y = t(a - x)$. The line $OP$ has equation $y = \frac{2}{t}x$. Eliminating $t$ from the tangent and perpendicular equations gives $y^2 = 2x(a - x)$ or $2x^2 + y^2 - 2ax = 0$.
Correct Answer: 2