Proving that two function coordinates of a parametric curve equals 1

Question

I am having difficulty with this question,

Note: This is not homework, It is from a practice test that I am using to study

Consider the curve:

$x(t) = \frac{1-t^2}{1+t^2}$ ; $y(t) = \frac{2t}{1+t^2} $

The question is to prove that $x^2(t) + y^2(t) = 1$ for all $t$

I have no clue where to start, I think it has something to do with the unit circle, but I'm not sure

Also, to find the slope of a parametric curve, Am i correct in assuming that you simply differentiate both of them, plug in the x and y, and use rise over run?

Thanks!

Related : http://math.stackexchange.com/questions/591062/ellipse-problem — lab bhattacharjee, Dec 04 '13 at 06:19
Thanks for the link, but I read through that and It seems like an unrelated problem — MathMajor, Dec 04 '13 at 06:21
Hmm, i'm not sure how I would apply the answer you gave over there to this question, any hints? — MathMajor, Dec 04 '13 at 06:24
you can put $t=\tan\theta$ or use $$\left(\frac{1-t^2}{1+t^2}\right)^2+ \left(\frac{2t}{1+t^2}\right)^2=1$$ — lab bhattacharjee, Dec 04 '13 at 06:25
Okay, I tried that, and this is what I got: $(1-t^2)^2 + 4t^2 = (1+t^2)^2$ — MathMajor, Dec 04 '13 at 06:28

score 1 · Accepted Answer · answered Dec 04 '13 at 06:25

1

Hint:

Take t=cos $\theta$. You will get x=$\cos2\theta$ and y=$\sin 2\theta$. Eliminate $\theta$ by using $$\cos^2 \theta+ \sin ^2\theta=1$$

answered Dec 04 '13 at 06:25

GTX OC

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Proving that two function coordinates of a parametric curve equals 1

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