Associating a focal length to an angle for a parabola

[rohanprabhu]

Homework Statement

If there is a line from the focus of a parabola such that it makes an angle $\theta$ with the x-axis. It intersects the parabola at a point 'P'. I want to find the focal length of the point P as a function of $\theta$.

The Attempt at a Solution

The equation of the parabola is given by $y^2 = 4ax$. The focus is given by $(a, 0)$.

The point where the line from the focus intersects the parabola is 'P' and the focal length of the point is given by 'l'. Then, the point 'P' is given by: $(a - l \cos ({\theta}), l \sin ({\theta}))$.

Then, to get, i substitute in the equation for a parabola and get a quadratic equation in 'l'. The problem i face now is in explaining in how exactly i get two solutions for 'l' [discriminant is dependent on certain factors and not necessarily 0]. For a given theta, it's pretty clear that i will be getting a certain length only. Then why is it so that i can get more than two solutions. What does the second solution signify?

 

[HallsofIvy]

The other solution is the where the other line at angle $\theta$ through (a, 0), the one with negative slope, intersects the parabola.