Finding the equation of a parabola, given the length of a portion of a focal chord, and the angle the chord makes with the parabola's axis

Find the equation of the parabola on a picture if $$|FL|=8$$ units and $$\angle KFO=60^o$$. $$F$$ is given as the focus of the parabola.

We know that this parabola passes through the point $$(0,0)$$, so if I can find a different point lying on this parabola I can find its equation. But I can't find this point.

How can I solve this problem?

Guide: The equation of this parabola is $$y^2=2px$$ for some real (and negative) $$p$$. Then $$F=({p\over 2},0)$$ and a line $$d$$ has an equation $$y=\sqrt{3}\left(x-{p\over 2}\right)$$
Solving the equation $$3\left(x-{p\over 2}\right)^2 = 2px$$you will get $$x$$ for $$L$$ (and $$K$$) and then you can calculate the $$y$$ for $$L$$. Use the fact $$LF =8$$ and the formula for the distance between two points and you will get a $$p$$.
• I like your answer, +1 from me--you do not give all details, which is appropriate for a question that shows no detailed work from the questioner. I could quibble a bit--I would use $f$, the distance from point $F$ to the origin, rather than $p$ in the equations, since it has a direct meaning in the diagram and avoids some fractions in the equations. I'll also edit your parentheses to make them look slightly better. But great job! – Rory Daulton Mar 23 at 12:09