# 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.

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

• Please state the problem explicitly, then explain using clear notation and a clear order of the introduced objects what was done. Two pictures, one with an unclear statement, one with a tiny raw picture and explanations with no obvious beginning and and can not substitute the needed requirements for a decent post. – dan_fulea Aug 22 at 18:42
• The question is what it is , I can't do anything about it , for the explanation though I will edit the post gladly – gucci Aug 22 at 19:03

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.