The tangent at (12,6) to a parabola intersects its directrix at (-1,2). The focus of the parabola lies on x-axis. number of such parabolas is

Question

I tried to proceed by assuming the focus to be $(a,0)$. Then I found the equation of tangent which came out to be $4x-13y+30=0$. I used the property of a parabola that the image of its focus about a tangent lies on its directrix. From here we can find the equation of the parabola in terms of $a$. I thought there would be only one such parabola. Also, is there a way to solve this question using some calculus properties of a parabola like that its second order derivative is a constant.

Answer 1

Let $F'$ be the reflection of focus $F$ about tangent $AB$, with $B=(-1,2)$ and $A=(12,6)$ the tangency point. As $AF=AF'$, then $\angle BF'A=90°$ and also $\angle BFA=90°$.

It follows that $F$ is an intersection between $x$-axis and the circle of diameter $AB$, and there are two of them (see figure below), at $(0,0)$ and $(11,0)$.