# Finding distance between focus and vertex of Parabola

A comet has a parabolic orbit with the sun at the focus, when the comet is 100 million km from the sun, the line joining the sun and the comet make an angle of 60 degrees, How close will the comet get to the sun?

I could only find the horizontal distance of comet from Sun that was $100 \cos60°\$million km.

Parabola is symmetric wrt positive x-axis

Let $a$ be the minimum distance of the comet from Sun, that is the distance between focus and vertex of the parabola. The equation of the parabola, with vertex at $(0,0)$, is then: $$x={1\over4a}y^2.$$ Substitute here the coordinates of the known position $C$ of the comet: $C=(a+50,50\sqrt3)$ (in units of $10^6\$km) and solve for $a$.
A more geometrical solution (see diagram below). Let $KH$ be the directrix of the parabola so that $CH=CS=100$. Triangle $CSH$ is equilateral, hence $KS={1\over2}CH=50$ and $VS={1\over2}KS=25$.