When given a cartesian function, told to plot in parametric, what is the best way to convert them? And why does it make sense to sum and square them?

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What is the relationship such that we are able to get trig functions out of equating them, importantly, how can I build that intuition?

I got $x= \sqrt{(5)^2-y^2}$ then subbed y with t to get parametric: $x= \pm\sqrt{(5)^2-t^2}$. This feels intuitive because this is the equation to get x value. I then do the same with x but it doesn't yield the results I expected. Reference question/context is below. I also attempted to graphed it out to visualize it. (am I keying it out in desmos wrongly..? link: https://www.desmos.com/calculator/xs93mhsmfd)

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