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$(X,Y)$ is uniform on the unit circle $(x^2 + y^2 = 1)$.

How do I calculate the conditional mean as a function of X = x?

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HINT: Conditional on $X=x$, $Y$ is distributed as $Y=-\sqrt{1-x^2}$ with probability $1/2$ and $Y=\sqrt{1-x^2}$ with probability $1/2.$ Does this make sense? So what is the conditional mean of $Y$? Does it depend on $x$?

The last question will help you with the regression. The regression function is the mean of $Y$ conditional on $X=x.$

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    This deals with the disk but the OP is asking about the circle, it seems. – 2017-02-05
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    @Did Originally they said unit disk before editing (and $x^2+y^2=1$, but I wrongly figured the equation was the mistake, not the words). Edited. – 2017-02-05
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    Perfect. // Unrelated: during the last 16 hours, this OP posted a total of 5 questions with zero context. One may feel this kind of systematic homework outsourcing is not what the site is about. – 2017-02-05
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    @Did Yes, I see they even asked a nearly identical question four hours before, got essentially the same response as I gave (only with the answer even less thinly veiled), did not reply, and then asked this one. – 2017-02-05