Chapter 9.7, Problem 27PPS

To determine

To find: The solution for the given system of the equation.

Answer to Problem 27PPS

The common region of the Figure 2 shows the solution of the inequalities.

Explanation of Solution

Given:

The given system of equation is,

  $\begin{array}{l}{x}^2+y^2\ge 36\\ x^2+9{\left(y+6\right)}^2\le 36\end{array}$

Calculation:

Consider the given equation are,

  $\begin{array}{l}{x}^2+y^2\ge 36\\ x^2+9{\left(y+6\right)}^2\le 36\end{array}$

The graph for the above two equation is shown in Figure 1

  

Figure 1

The inequality $x^2+y^2\ge 36$ is greater than the equal to sign, thus, the region exterior of the circle and the boundary of the circle and that satisfy the inequality.

The inequality $x^2+9{\left(y+6\right)}^2\le 36$ is less than equal to sign. The point (0, 0) from the exterior of the eclipse does not satisfy the inequality. Thus, the region interior of the ellipse and on the boundary of the ellipse will satisfy the inequality.

Thus, the solution for the inequalities is shown in Figure 2

  

Figure 2

The common region represents the solution of the system.