# Proof of a focus point on parabola and tangent line equal angles

1. Oct 12, 2009

### lax1113

1. The problem statement, all variables and given/known data
Prove that a vertical line and a line going from a point on a parabola to the focus of the parabola form equal angles with the tangent line of the point on the parabola.

2. Relevant equations
Focus = 1/4a (maybe relevant)

3. The attempt at a solution
I know how to prove that the triangle from the vertical line, midpoint of Focus point to an arbitrary line and the point on the parabola is equal to a triangle that goes from focus point to point on parabola to midpoint.

However, I have no clue how to show that these two angles are the same. I can find the slope of each line, obviously, but where to go from here?

2. Oct 13, 2009

### Mentallic

Don't worry about any triangles, simply do as the question asks.
Take any arbitrary point $P(2ap,ap^2)$ on the parabola $x^2=4ay$ where $(0,a)$ is the focus. Now, find the gradient of the tangent to the parabola which touches at P, also take the gradient of the line connecting the focus and the point P. Now find the angle between these 2 lines with the equation:

$$tan\theta=\left |\frac{m_1-m_2}{1+m_1m_2} \right |$$

Now take the gradient of a vertical line which is $1/0$ (don't worry that it is undefined, with the tan function that just means $\theta=\pi/2$) and now show the angle between that tangent line and the vertical line is the same.