Question

The circle on focal radii of a parabola as diameter touches the

• A. axis
• B. directrix
• C. tangent at vertex
• D. None of these

Answer 1

Answer: (C)

tangent at vertex

Let the parabola be $y^2=4ax$.

Let $P(a{t}^2,2at)$ be any point on the parabola.

Then $SP$ is focal radii of the parabola.

The equation of a circle with $SP$ as diameter is

$(x-a)(x-a{t}^2)+(y-0)(y-2at)=0$

It meets y-axis at $x=0$

$\therefore y^2-2aty+a^2{t}^2=0\Rightarrow {(y-at)}^2=0$

$\Rightarrow$ y-axis meets the circle only at one point

Therefore, the circle touches the tangent at the vertex.