Thread: Proof involving hyberbola

Prove that the line touches the hyperbola if .

How do I start?

Edit: Sorry for the error in the question.

Check the wording of the question. What you've written isn't a line.

Corrected the question.

Again have you copied the question correctly. That is an ellipse not a hyperbola.

biffboy, thanks for your patience. Error again and corrected again.

Make x the subject of the line equation and substitute this into the hyperbola equation. Rearrange this as a quadratic in y. For the line to be a tangent we require this quadratic to have repeated roots. Applying this condition should produce the result stated.

So, do I substitute the value of y in the line equation with the value of y with respect to x?

In that case, the equation becomes,



So the line equation will become



I tried to simplify the equation and it is becoming messy. Am I doing things right?

(Sorry for the small image, laTEX is pain and I do not know how to put the special characters here. Tried to use an online latex editor but there seems to be some problem)

Replace y in the hyperbola equation with (p-xcosa)/sina

This doesn't seem to be leading anywhere. Is this what I am supposed to do?

Rearrange your last line to get a quadratic in y. You then want this quadratic to have a repeated root.

What you have substituted for x should be p/cosa-(sina/cosa)y

So I got the equation



Is this correct?

I can't see what's happened there. Looking back your last line of post 9 was correct so look at my post 10 again

If you want one last fresh start try this. I'll write A instead of alpha. Line can be written y=-(cosA/sinA)x+p/sinA
This is y=mx+c with m=-cosA/sinA and c=p/sinA
Substitute y=mx+c into the hyperbola. So x^2/a^2-(mx+c)^2/b^2=1
Multiply throughout by a^2b^2 and remove the brackets and then rearrange as a quadratic in x
Apply the condition for this to have repeated roots
After some symplifying this gives b^2+c^2-(a^2m^2)=0
Put back in what m and c stood for to get the required result

OK, here's my attempt:



Multiplying both the terms by a²b²,



Rearranging, I get



This is of the form ax²+bx+c=0

What do I do now? I don't know how to make it have repeated roots.

I hope I am on the right track.