Question

Which equation represents the polar form of x² + (y + 4)² = 16?

Answer 1

$x^2+y^2=r^2\hspace{5em}x=r\cos(\theta )\hspace{5em}y=r\sin(\theta ) \\\\[-0.35em] ~\dotfill\\\\ x^2+(y+4)^2=16\implies x^2+\stackrel{binomial~expansion}{y^2+8y+16}=16 \\\\\\ \underset{x^2+y^2}{r^2} + 8(~\underset{8y}{r\sin(\theta )}~)=16-16\implies r^2+8r\sin(\theta)=0 \\\\\\ r^2=-8r\sin(\theta )\implies \cfrac{r^2}{r}=-8\sin(\theta )\implies r=-8\sin(\theta )$