B Prove that tangents to the focal cord of parabola....

[Vriska]

Prove that tangents to the focal cord of parabola are perpendicular using the reflection property of parabola ( A ray of light striking parallel to the focal plane goes through the focus, and a ray of light going through the focus goes parallel)

I don't know whether this is solvable with just this much detail, just something I've been thinking about, looks doable but I've not been able to get anywhere. The geometry is quite tangled.

 

[andrewkirk]

The geometry is reasonably simple as long as one approaches it from the correct perspective.

Consider a parabola in the number plane whose axis of symmetry is the y-axis and the apex is the origin. Now consider two rays that travel vertically down and are reflected along the focal chord, which is horizontal. One of the rays descends at abscissa $a$ and the other at abscissa $-a$, where the value of $a$ depends on the shape of the parabola.

Consider the ray that comes down at abscissa $a$. What is the angle between its incident ray and its reflection? What does that enable us to say about the angle between the focal chord (along which the reflected ray travels) and the tangent to the parabola at the point of reflection?

 

[Vriska]

The geometry is reasonably simple as long as one approaches it from the correct perspective.

Consider a parabola in the number plane whose axis of symmetry is the y-axis and the apex is the origin. Now consider two rays that travel vertically down and are reflected along the focal chord, which is horizontal. One of the rays descends at abscissa $a$ and the other at abscissa $-a$, where the value of $a$ depends on the shape of the parabola.

Consider the ray that comes down at abscissa $a$. What is the angle between its incident ray and its reflection? What does that enable us to say about the angle between the focal chord (along which the reflected ray travels) and the tangent to the parabola at the point of reflection?

The angle between the reflected and incident is 90 - the angle of the tangent wrt the perpendicular is 45. I'm not sure what to do after this, I'm still finding the geometry tangled

 

[andrewkirk]

OK, now what about the tangent at the point of reflection of the ray that comes down at abscissa $-a$.

What can we then say about the angle between that and the other tangent?

 

[Vriska]

OK, now what about the tangent at the point of reflection of the ray that comes down at abscissa $-a$.

What can we then say about the angle between that and the other tangent?

One is 45 the other is - 45. Angle between them is 90

 

[andrewkirk]

You're welcome.

 

[Vriska]

You're welcome.

? noo , not this, the question was to prove *any* focal chord through the focus (not only latus rectum) has tangents that are perpendicular. I can do it with analytical geometry, but i want to do it with the reflection thing.

 

[andrewkirk]

The OP said *the*, not *any*.

But in any case, the argument needs only minor adjuistment to apply it to any focal chord. Just consider the two rays heading parallel to the parabola axis after reflection at either end of a given focal chord. Label the angle of incidence for one of those rays $a$ and that for the other as $b$. Using rules of reflection and completing the triangle formed by the focal chord and the two tangents we see that the angle between the tangents is right.

 

[Vriska]

The OP said *the*, not *any*.

But in any case, the argument needs only minor adjuistment to apply it to any focal chord. Just consider the two rays heading parallel to the parabola axis after reflection at either end of a given focal chord. Label the angle of incidence for one of those rays $a$ and that for the other as $b$. Using rules of reflection and completing the triangle formed by the focal chord and the two tangents we see that the angle between the tangents is right.

Ah thank you so much, the triangle thing was brilliant - and simple.