# what's the algebra (if any) behind converting f(x) for a circle to a parametric equation

I'm sure there has to be some algebra behind it. My problem called to covert $$(x - 2)^2 + (y - 9)^2 = 4$$ if $x = 2 + 2cos(t)$ then $y = ?$

I know the answer is $9 + 2\sin(t)$ but I simply got that by looking at the formula in my math book. $$x = h + r cos(t), y = k + r sin(t), 0\leq t \leq2\pi$$. That's all the book really gives me on this.

But I'm pretty sure if this were to show up on my final I'd have to so some algebra for it. Can anyone please enlighten me a little? Is there more to it or is that it as far as showing work goes?

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$$\left(\frac{x-2}2\right)^2+\left(\frac{y-9}2\right)^2=1$$

We know, $\cos^2t+\sin^2t=1$

If one of the term above the left say, $\frac{x-2}2=\cos t$, the other will be $\sin t$

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Geometrically, $x = 2 + 2\cos t$ has meaning. Given a point $P = (x,y)$ on your circle with centre $O$, $t$ is the angle of the the line $OP$ with a line parallel to the x-axis.

Draw a figure and find out why such a substitution is natural!

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