A circle through a point on a parabola and the foot of the perpendicular to the directrix has its center on the tangent. Prove it contains the focus.

Question

Here I first drew a parabola with equation $y^2=4ax$ then according to the question I drew a tangent at point $P(at^2,2at)$ which passes through the foot of directrix ie: $T(-a,0)$ . Now I drew a circle with centre $C(h,k)$ which lies on the tangent at $P$ and also the circle passes through the the point $P$ . Now we have to prove that the circle always passes through the focus ie: $S(a,0)$

I did the following calculation

I am going wrong somewhere which I am not being able to identify. I thoroughly checked my calculation again and again but didnt find a mistake but still I am wrong in some place . It would be greatly appreciated if someone could help me find my mistake

Answer 1

Your diagram is not correct, because you misinterpreted the text (which is indeed rather obscure). The text should read as follows:

A circle is drawn through any point $P$ on the parabola $y^2=4ax$ and through the perpendicular projection of $P$ on the directrix. The centre of the circle lies on the tangent at $P$. Prove the circle always passes through the focus.

I'd suggest you to check with GeoGebra if the diagram is correct, before starting to prove something.

The solution of this problem, by the way, can be carried out with no calculations at all, by using the properties of the parabola I reported in my answer to your previous question.