Chapter 12, Problem 2CR

a.

To determine

To graph:The equation of circle and parabola.

Answer to Problem 2CR

The graph of the equation $x^2+y^2=100$ is shown in Figure1 and $y=3x^2$ is shown in Figure2.

Explanation of Solution

Given information:

The given equation of circle and parabola are,

  $\begin{array}{c}{x}^2+y^2=100\\ y=3x^2\end{array}$

Calculation:

Draw the graph of the equation of the circle.

  

Figure1

Draw the graph of the equation of the parabola.

  

Figure2

Therefore, the graph of the equation $x^2+y^2=100$ is shown in Figure1 and $y=3x^2$ is shown in Figure2.

b.

To determine

To find: The system of equations.

Answer to Problem 2CR

The system of equations of two factors is $y=9\cdot 835,-10\cdot 16$ .

Explanation of Solution

Given information:

The given equations are,

  $x^2+y^2=100$ …… $\left(1\right)$

  $y=3x^2$ …… $\left(2\right)$

Calculation:

For solving these two equations, we will put equation $\left(2\right)$ in equation $\left(1\right)$ .

So, we will put $x^2=\frac{y}{3}$ in equation $\left(1\right)$ .

Therefore

  $\begin{array}{c}{x}^2+y^2=100\\ \frac{y}{3}+y^2=100\\ y^2+\frac{y}{3}-100=0\\ 3y^2+y-300=0\\ \end{array}$

So, we will find the factor of $y$ .

  $\begin{array}{c}y=\frac{-1\pm \sqrt{\left(1^2\right)-4\times 3\left(-300\right)}}{2\times 3}\\ =\frac{-1\pm \sqrt{1-\left(-3600\right)}}{6}\\ =\frac{-1\pm \sqrt{1+3600}}{6}\\ =\frac{-1\pm \sqrt{3601}}{6}\\ y=\frac{-1\pm 60\cdot 01}{6}\end{array}$

So, $y$ will have two factors

  $\begin{array}{c}y=\frac{-1+60\cdot 01}{6}\\ =\frac{59\cdot 01}{6}\\ y=9\cdot 835\end{array}$

Or

  $\begin{array}{c}y=\frac{-1-60\cdot 01}{6}\\ =\frac{-61\cdot 01}{6}\\ y=-10\cdot 16\end{array}$

Therefore, the system of equations two factor is $y=9\cdot 835,-10\cdot 16$ .

c.

To determine

To find:The circle and parabola intersect

Answer to Problem 2CR

The intersection of given parabola and circle are $\left(-1\cdot 81,9\cdot 835\right)$ and $\left(-1\cdot 81,9\cdot 835\right)$ .

Explanation of Solution

Given information:

The given values are $y=9\cdot 835,-10\cdot 16$ .

Calculation:

  $y=-10\cdot 16$ Is not possible as it not on circle rather it is out of circle area.so, it can’t be a point of intersection. So, $y=9\cdot 835$ .

We will put $y=9\cdot 835$ in equation $\left(2\right)$ .

  $\begin{array}{c}9\cdot 835=3x^2\\ x^2=\frac{9\cdot 825}{3}\\ x^2=3\cdot 2783\\ x=\pm \sqrt{3\cdot 2783}\\ x=\pm 1\cdot 81\end{array}$

Therefore, the intersection of given parabola and circle are $\left(-1\cdot 81,9\cdot 835\right)$ and $\left(-1\cdot 81,9\cdot 835\right)$ .