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

[lax1113]

Homework Statement

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.

Homework Equations

Focus = 1/4a (maybe relevant)

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?

 

[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.

 

[Architeuthis Dux]

I would approach this problem by first understanding the properties of a parabola and its focus. The focus of a parabola is a fixed point that is equidistant from the directrix and the vertex. This means that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix.

Next, I would consider the definition of a tangent line, which is a line that intersects a curve at exactly one point and is perpendicular to the curve at that point. This means that the tangent line at any point on the parabola is perpendicular to the line connecting that point to the focus.

Now, let us consider the given scenario. We have a vertical line intersecting the parabola at a point, and a line connecting that point to the focus of the parabola. We want to show that these two lines form equal angles with the tangent line at the point of intersection.

To begin, we can assume that the parabola has the equation y = ax^2, where a is a constant. We can also assume that the point of intersection of the vertical line and the parabola is (x0, y0).

Using this information, we can find the coordinates of the focus of the parabola. Since the focus is located at a distance of 1/4a from the vertex, and the vertex is at (0,0), the focus will be at (0,1/4a).

Now, let us consider the slope of the tangent line at the point (x0, y0). We can find this by taking the derivative of the parabola equation, which gives us the slope at any point on the parabola as 2ax0.

Next, we can find the slope of the line connecting (x0, y0) and (0,1/4a), which is given by (y0 - 1/4a)/(x0 - 0). Simplifying this, we get (4ay0 - 1)/(4ax0).

Since we want to show that the angles formed by the tangent line and the two given lines are equal, we can compare their slopes. We can see that the slopes of both the lines are equal when (4ay0 - 1)/(4ax0) = 2ax0.

Solving for x0, we get x0 = 1