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 [itex]\theta[/itex] 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 [itex]\theta[/itex].

The Attempt at a Solution

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

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: [itex](a - l \cos ({\theta}), l \sin ({\theta}))[/itex].

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 [itex]\theta[/itex] through (a, 0), the one with negative slope, intersects the parabola.

 

[Architeuthis Dux]

I would approach this problem by first clarifying the definitions and assumptions being made. The focal length of a parabola is typically defined as the distance from the vertex (the point where the parabola changes direction) to the focus. However, in this problem, the focus is defined as the point (a,0) which is not necessarily the vertex. Additionally, the equation y^2 = 4ax that is being used assumes that the parabola is oriented with its axis along the x-axis, which may not always be the case.

With these clarifications in mind, we can proceed with the solution. The focal length of a point P on a parabola is given by the distance from P to the focus. Using the distance formula, we can write this as:

l = √[(a - x)^2 + y^2]

where (x,y) is the coordinates of point P. Substituting in the coordinates given in the problem, we get:

l = √[(a - a cosθ)^2 + (a sinθ)^2]

= √[a^2(cos^2θ - 2acosθ + 1) + a^2sin^2θ]

= √[a^2(cos^2θ + sin^2θ) - 2a^2cosθ + a^2]

= √[a^2(1 - cosθ) + a^2]

= √[2a^2(1 - cosθ)]

= a√[2(1 - cosθ)]

This gives us the focal length l as a function of the angle θ. However, this solution assumes that the point P lies on the parabola. If we allow P to be any point in the plane, then we can get multiple solutions for l. This is because the equation y^2 = 4ax represents a family of parabolas, each with a different vertex and focus. So, for a given angle θ, there may be multiple parabolas that intersect the line from the focus at point P, resulting in multiple solutions for l.

In conclusion, the second solution for l signifies that there are multiple parabolas that satisfy the given conditions, and thus multiple possible focal lengths for the point P. This is due to the fact that the equation y^2 = 4ax represents a family of parabolas