Find the Cartesian equation of the curve

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Homework Statement
This is a past paper question...see attached (my interest is on the highlighted) I used a different approach thus your insight would be great. I always make it a point to try out the questions before checking what the mark scheme offers...
Relevant Equations
parametric equations.
1671348574228.png

Find ms solution;

1671348656844.png


My approach;

##xt=t^2+2## and ##yt=t^2-2##

##xt-2=t^2## and ##yt+2=t^2##

##⇒xt-2=yt+2##

##xt-yt=4##

##t(x-y)=4##

##t=\dfrac{4}{x-y}##

We know that;

##x+y=2t##

##x+y=2⋅\dfrac{4}{x-y}##

##(x-y)(x+y)=8##

##x^2-y^2=8##

Your insight is welcome...rather asking if this approach would be correct.
 
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x+y=2t
x-y=\frac{4}{t}
(x+y)(x-y)=8
 
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Likes pasmith, topsquark, SammyS and 1 other person
Writing <br /> \begin{split}<br /> x &amp;= t + \frac{2}{t} = 2\sqrt{2}\operatorname{sgn}(t)\cosh(\ln(|t|/\sqrt{2})) \\<br /> y &amp;= t - \frac{2}{t} = 2\sqrt{2}\operatorname{sgn}(t)\sinh(\ln(|t|/\sqrt{2}))\end{split} shows that the curve is a hyperbola, and basic identities then give x^2 - y^2 = (2\sqrt{2})^2 = 8.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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