Bivariate pdf of $X$ and $Y$

Question

The bivariate pdf of $X$ and $Y$ is the following:

$f(x,y) = K$, if $y>x$ and $x^2 + y^2 \leq 1\;\;(f(x,y)= 0$ otherwise)

Find the constant $K$, the marginal pdf of $Y$, and the conditional pdf of $X$ given $Y=0$

I know that to find $K$, we must integrate and set equal to $1$. However, I am having trouble determining the bounds. I would assume, because it is a circle with a line through it diagonally, the bounds may involve $1/\sqrt2$ or perhaps $\sqrt{(1-x^2)}$...

Answer 1

The support is $$\{(X,Y): 1/\sqrt 2\leq Y\lt 1 ~, {-\sqrt{1-Y^2}}\leq X\leq \sqrt{1-Y^2}\}\cup\{(X,Y):-1/\sqrt 2\leq Y< 1/\sqrt 2~, {-\sqrt{1-Y^2}}\leq X\lt Y\}$$

This is half the unit circle, so you can find $K$ without that.

However, it may be of help for finding the marginal distributions.