Chapter 7, Problem 17CT

To determine

To sketch: the graph of the given solution set of the system of in equalities.

Answer to Problem 17CT

The points of intersections of the curve and the line are $\left(2,-4\right),\left(2,\pm 4\sqrt{2}\right)\text{and }\left(\pm 2\sqrt{5},-4\right)$ .

Explanation of Solution

Given:

The given solution is

  $\{\begin{array}{l}{x}^2+y^2\le 36\\ x\ge 2\\ y\le -4\end{array}$

Calculation:

Consider the system of inequalities

  $\{\begin{array}{l}{x}^2+y^2\le \mathrm{36...............}(1)\\ x\ge \mathrm{2..........................}(2)\\ y\le -\mathrm{4..........................}(3)\end{array}$

Consider the corresponding equations.

  $\begin{array}{l}{x}^2+y^2=\mathrm{36...............}(4)\\ x=\mathrm{2..........................}(5)\\ y=-\mathrm{4..........................}(6)\end{array}$

The points of intersection of the curve (4) and the line (5), and (6) are given by

  $\begin{array}{c}{\left(2\right)}^2+y^2=36\\ y^2=32\\ y=\pm 4\sqrt{2}\end{array}$

And

  $\begin{array}{c}{x}^2+{\left(-4\right)}^2=36\\ x^2=20\\ x=\pm 2\sqrt{5}\end{array}$

Thus the points of intersections of the curve and the line are $\left(2,\pm 4\sqrt{2}\right)\text{and }\left(\pm 2\sqrt{5},-4\right)$ .

Further another point of intersection is $\left(2,-4\right)$ .

The graph is shown below.

  

The point U represents $\left(2\sqrt{5},-4\right)$ and the point M represents $\left(2,4\sqrt{2}\right)$ .