Proving the Reflection of Light Ray from Parabolic Mirror on x-Axis

In summary, the conversation is discussing how to prove that a light ray emitted from the focus of a parabolic mirror will be reflected parallel to the x-axis. The conversation involves considering the parabola, diagram, and gradients of certain lines to show that the triangle formed is similar. The conversation also touches on using angles and the tangent function to further prove this statement.
  • #1
PotatoSalad
11
0
I'm hopelessly stuck on this question. Any help will be greatly appreciated.

Prove that if we have a parabolic mirror with focus at F and axis of symmetry the x-axis, then a light ray emmited from F will be reflected parallel to the x-axis.

To prove this consider the parabola y^2=4px (where the focus is at the point (p,0) and the directrix is the line x=-p) and the diagram (below) where N is the point on the directrix which is nearest to P and AP is a tangent to the parabola. Consider the gradients of FN and AP. Use this information to show that the triangle PAF is similar to PAN.


http://img92.imageshack.us/img92/3983/parabolayp5.jpg

I have managed to show that the gradient of the line PA is (2p/y) and FN is (-y/2p) so those 2 lines are perpendicular. But this is about the only progress I have made with the question.

Thanks for your help.

EDIT:
Sorry, just noticed my diagram is slightly wrong. The point A should be on the y-axis and should also intersect with FN.
 
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  • #2
Extend line FP beyond the parabola and extend line NP to the right. You should be able to see that, since reflections have angles of incoming and outgoing rays the same, the angle NP extended makes with the tangent (the angle of the incoming ray) is equal to the angle FP (the outgoing ray) makes with the tangent. Then "vertical angles" shows that the angle FP makes with the tangent is the same as is the same as FP extended makes with the tangent- which is the same that the incoming ray, NP extended, makes with the tangent. But then that angel NP extended makes with FP extended is TWICE the angle NP extended makes with the tangent.

And the angle FP makes with NP extended is ("corresponding angles in parallel lines") the same as FP makes with the axis of the parabola: twice the angle NP extended makes with the tangent. Now use the fact that the tangent of that angle is the slope of line FP and that
[tex]tan(2\theta)= \frac{2tan(\theta)}{1- tan^2(\theta)}[/tex]
 
  • #3
HallsofIvy said:
Then "vertical angles" shows that the angle FP makes with the tangent is the same as is the same as FP extended makes with the tangent- which is the same that the incoming ray, NP extended, makes with the tangent.
I don't see how you have come to this conclusion.Or are you just trying to say that for it to work those angles must be equal but you have not yet proved it?

HallsofIvy said:
But then that angel NP extended makes with FP extended is TWICE the angle NP extended makes with the tangent.
Again, are you just saying that this must be true (in which case I understand why that is), or are you trying to claim that you have already proved it is true?

HallsofIvy said:
And the angle FP makes with NP extended is ("corresponding angles in parallel lines") the same as FP makes with the axis of the parabola: twice the angle NP extended makes with the tangent.
HallsofIvy said:
Now use the fact that the tangent of that angle is the slope of line FP and that
[tex]tan(2\theta)= \frac{2tan(\theta)}{1- tan^2(\theta)}[/tex]

Please explain why the bold part is true. Sorry I am struggling so much with this question, we only got introduced to conic sections last week so there are still a lot of things I have yet to get my head around with it.

Thanks very much for your help.
 

Related to Proving the Reflection of Light Ray from Parabolic Mirror on x-Axis

1. How does a parabolic mirror reflect light rays onto the x-axis?

A parabolic mirror is a curved reflective surface in the shape of a parabola. When light rays hit the mirror, they are reflected off the surface and converge at a single focal point on the x-axis. This is due to the unique shape of the parabolic mirror, which causes all incoming parallel light rays to reflect and converge at the focal point.

2. What is the principle behind the reflection of light rays from a parabolic mirror onto the x-axis?

The principle behind the reflection of light rays from a parabolic mirror onto the x-axis is the law of reflection, which states that the angle of incidence (the angle at which the light ray hits the mirror) is equal to the angle of reflection (the angle at which the light ray bounces off the mirror). This means that all incoming light rays will be reflected in a predictable manner, resulting in a focal point on the x-axis.

3. How can we prove the reflection of light rays from a parabolic mirror onto the x-axis?

To prove the reflection of light rays from a parabolic mirror onto the x-axis, we can use the principle of symmetry. This means that the angle of incidence and the angle of reflection will be equal on opposite sides of the parabolic mirror. By measuring and comparing the angles of incidence and reflection on both sides of the mirror, we can demonstrate the reflection of light rays onto the x-axis.

4. What factors can affect the reflection of light rays from a parabolic mirror onto the x-axis?

The main factor that can affect the reflection of light rays from a parabolic mirror onto the x-axis is the shape and curvature of the mirror. If the mirror is not perfectly parabolic, the reflected light rays may not converge at the focal point on the x-axis. Other factors such as the quality and smoothness of the reflective surface can also impact the accuracy of the reflection.

5. How is the reflection of light rays from a parabolic mirror onto the x-axis used in practical applications?

The reflection of light rays from a parabolic mirror onto the x-axis has many practical applications, including in telescopes and satellite dishes. These mirrors are used to focus and gather light from distant objects and direct them onto the x-axis, allowing for clearer and more detailed observations. Parabolic mirrors are also used in solar power plants, where they concentrate sunlight onto a single point on the x-axis to generate heat and electricity.

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