Find the Cartesian equation of curve defined parametrically

[Monkeyseat]

Hi,

Question

A curve is define by the parametric equations x = (t^2) + (2/t), y = (t^2) - (2/t).

Verify that the Cartesian equation of the curve is (x + y)(x - y)^2 = k, stating the value of the constant k.

Working

I don't really have any working as I can't seem to get anywhere with this. I realise I need to eliminate the parameter t, but I'm not sure how I should go about it.

I thought about putting the expressions for x and y over the same denominator:

x = ((t^4) + 2t)/(t^2) and y = ((t^4) - 2t)/(t^2)

But I'm not sure where to go from there (or if that is even the 'correct' way to go about this question). Alternatively, I tried this:

x = (t^2) + (2/t) and y = (t^2) - (2/t).
ty = (t^3) -2 and tx = (t^3) + 2
ty - (t^3) = -2 and tx - (t^3) = 2

But again, I don't know how to advance from there.

Any help is greatly appreciated.

Thanks.

 

[skeeter]

A curve is define by the parametric equations x = (t^2) + (2/t), y = (t^2) - (2/t).

Verify that the Cartesian equation of the curve is (x + y)(x - y)^2 = k, stating the value of the constant k.

why make this more difficult than it needs top be ? ... just follow the given directions.

\(\displaystyle x+y = 2t^2\)

\(\displaystyle x-y = \frac{4}{t}\)

\(\displaystyle (x-y)^2 = \frac{16}{t^2}\)

\(\displaystyle (x+y)(x-y)^2 = 2t^2\left(\frac{16}{t^2}\right) = 32\)

you're done.

 

[Monkeyseat]

I didn't think of doing that. It's certainly much simpler than what I was trying.

Thanks for helping.