[tex]\bf \begin{cases}
x=rcos(\theta )\\
y=rsin(\theta )\\
x^2+y^2=r^2\implies y^2=r^2-x^2
\end{cases}
\\\\\\
sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)\\\\
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y=\sqrt{x}\implies y^2=x\implies r^2-x^2=x\implies \cfrac{r^2-x^2}{x}=1
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\cfrac{r^2}{x}-\cfrac{x^2}{x}=1\implies \cfrac{r^2}{x}-x=1\implies \cfrac{r^2}{rcos(\theta )}-rcos(\theta )=1[/tex]
[tex]\bf \cfrac{r}{cos(\theta )}-rcos(\theta )=1\implies r\left[ \cfrac{1}{cos(\theta )}-cos(\theta ) \right]=1
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r\left[ \cfrac{1-cos^2(\theta )}{cos(\theta )} \right]=1\implies r\left[ \cfrac{sin^2(\theta )}{cos(\theta )} \right]=1\implies r=\cfrac{1}{\frac{sin^2(\theta )}{cos(\theta )}}
\\\\\\
r=\cfrac{cos(\theta )}{sin^2(\theta )}[/tex]
