Question

2. (a) Graph the circle $x^2+y^2=9$and the parabola $y=3x^2$

(b) Solve the system of equations:$x^2+y^2=9,y=3x^2$

(c) Where do the circle and the parabola intersect?

Short answer

(a) The graph of circle and parabola is shown in the figure.

(b) The solution of system of equation is $\left(-1.811,9.8\right),\left(1.811,9.8\right)$.

(c) The parabola and circle intersect at$\left(-1.811,9.8\right),\left(1.811,9.8\right)$

Step-by-step solution

Step 1. Given Information

Given circle $x^2+y^2=100$with centre $\left(0,0\right)$and radius $10.$

and equation of parabola$y=3x^2\Rightarrow x^2=\frac{y}{3}$

Part (a) Step 1. Graph of the circle and parabola

Part (b) Step 1. Solution of system of equation

$$\begin{gathered} x^2+y^2=100-----\left(1\right) \\ y=3x^2-----\left(2\right) \end{gathered}$$

Substitute $y=3x^2$into $x^2+y^2=100$

role="math" localid="1646975709876" $$\begin{gathered} x^2+{\left(3x^2\right)}^2=100 \\ {x}^2+9x^4=100 \\ 9x^4+x^2-100=0 \\ 9y^2+y-100=0 \end{gathered}$$

Part (b) Step 2. Solving equation

$$\begin{gathered} 9x^2+x-100=0 \\ x=\frac{-1\pm \sqrt{1-4\times 9\times -100}}{2\times 9} \\ x=\frac{-1\pm \sqrt{3601}}{18} \\ x=\frac{-1\pm 60.0083}{18} \\ x=\frac{-1+60.0083}{18},\frac{-1-60.0083}{18} \\ x=3.2782,-3.38935 \end{gathered}$$

Part (b) Step 3. Finding the value of y.

Substitute $x=3.2782$in $y=3x^2$

$$\begin{gathered} y=3{\left(3.2782\right)}^2 \\ y=32.23 \end{gathered}$$

Part (c) Step 1. Intersecting points

From the graph we can see that the parabola and circle intersect at$\left(-1.811,9.8\right),\left(1.811,9.8\right)$