Question

if the xy-plane, the graph of y=x^2 and the circle with center (0,1) and radious 3 have how many points of intersection?

Answer 1

$\bf \textit{equation of a circle}\\\\ (x-{{ h}})^2+(y-{{ k}})^2={{ r}}^2 \qquad \begin{cases} center\ ({{ h}},{{ k}})\\ radius={{ r}}\\ ----------\\ center\ (0,1)\to \begin{cases} h=0\\ k=1 \end{cases}\\ r=3 \end{cases} \\\\\\ (x-0)^2+(y-1)^2=3^2\implies x^2+(y-1)^2=9 \\\\\\ (y-1)^2=9-x^2\implies y=\sqrt{9-x^2}+1\\\\ -----------------------------\\\\ \textit{where do the parabola and circle intersect? set them to equal}$

$\bf \begin{cases} y=x^2\\\\ y=\sqrt{9-x^2}+1 \end{cases}\qquad y=y\implies x^2=\sqrt{9-x^2}+1$

solve for "x"