Proof that Dandelin sphere is focus of the parabola

Question

Suppose a plane intersects a right-circular cone in a parabola with vertex at V: Suppose that a sphere is inscribed between the cone and the plane as in the previous exercises and is tangent to the plane of the parabola at point F:

Show that the chord to the parabola through F that is perpendicular to F V has length equal to that of the latus rectum of the parabola.Therefore, F is the focus of the parabola.

I know that the semi latus rectum of parabola should be twice as long compared to line from focus to vertex but i couldn't come up with a way proves that.

I'm a chemical engineer graduate, trying to learn math from scratch.

Edit:
Solution of previous exercise, a classic proof:

Since tangents from a outer point to a sphere have same length.

$PB=PF_2$ and $PA=PF_1$

$PB+PA=PF_2+PF_1= AB = \ \text{constant}$

proving $F_1$ and $F_2$ are focal points.

Answer 1

$$ PF=PM=BT=2BV=2VF. $$

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