The circles on the focal length of a parabola as diameter always touches

The tangent at the vertex

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The axis

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The directrix

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Latus rectum

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Solution

The correct option is A The tangent at the vertex
The circle on $S P$ as diameter is
$(x - a t_{1}^{2}) (x - a) + (y - 2 a t_{1}) y = 0 \Rightarrow x^{2} + y^{2} - a x (t_{1}^{2} + 1) - 2 a y t_{1} + a^{2} t_{1}^{2} = 0$

The $y$ intercept of the circle will be
$2 \sqrt{f^{2} - c} = \sqrt{a^{2} t_{1}^{2} - a^{2} t_{1}^{2}} = 0$
$∵ y$ intercept made by circle is $0$ hence circle will touch the $y -$ axis which is tangent at the vertex.

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