Find the rectangular equation and orientation of the curve

Question

The given parametric equations are $$x=t^2-2$$ $$y=5-2t$$ solving the $x$ equation I get $t=\pm\sqrt{x+2}$ then plugging that value into the $y$ equation gives me $$y=5-2\sqrt{x+2}$$ from this I can sketch the curve with is orientated counter-clockwise to the right. When I checked my answer the final equation should be $(y-5)^2=4x+8$. So am I meant to manipulate the equation I have now to get that or is it something else?

Answer 1

$t$ should be $\pm \sqrt{x+1}$ and the final answer should be $y = 5\pm 2\sqrt{x+1} $

which is the same way of saying ${(y-5})^2 = 4x+4$

This mistake you have done is, you have only included the negative solution of $t$ so you only get 1 part of the curve, when you include the other part as well(the + solution) you will get the complete curve.

I hope this helps.