Chapter 10.9, Problem 44E

To determine

To find: find the coordinates of all vertices and determine whether the solution set bounds or not.

Answer to Problem 44E

The parabola do not intersect

Explanation of Solution

Given:

  $\{\begin{array}{l}{x}^2-y\ge 0\\ x+y<6\\ x-y<6\end{array}$

Calculation:

The given equation is

  $\begin{array}{l}{x}^2-y\ge 0\\ x+y<6\\ x-y<6\end{array}$

Here, it is given that

  $\begin{array}{l}{x}^2-y\ge 0\\ x+y<6\\ x-y<6\end{array}$

Then, we take

  $\begin{array}{l}{x}^2-y\ge \mathrm{0........}\left(1\right)\\ x+y<\mathrm{6........}\left(2\right)\\ x-y<\mathrm{6........}\left(3\right)\end{array}$

From equation $\left(1\right)$ , $x^2=y$ is a parabola, which has vertex $\left(0,0\right)$

From equation $\left(2\right)$ , $\left(2\right)$ and $\left(3\right)$ are both straight lines

  $\begin{array}{l}x+y=6\\ \frac{x}{6}+\frac{y}{6}=1\end{array}$

Then, line $\left(2\right)$ intersects the axes at $\left(6,0\right)$ and $\left(0,6\right)$

And for equation $\left(3\right)$ .

  $\begin{array}{l}x-y=6\\ \frac{x}{6}+\frac{y}{-6}=1\end{array}$

Then, line $\left(3\right)$ intersects the axes at $\left(6,0\right)$ and $\left(0,-6\right)$

To find the intersection point of the lines $\left(2\right)$ and $\left(3\right)$ .

Solve $\left(2\right)$ and $\left(3\right)$

We get $x=6,y=0$

Then, $\left(6,0\right)$ is the intersection point of both lines

From equation $\left(2\right)$ , we have

  $y=6-x$

Substituting into equation $\left(1\right)$ , we have

  $\begin{array}{l}{x}^2+6x+x=0\\ x^2+x-6=0\\ \left(x+3\right)\left(x-2\right)=0\end{array}$

  $x=-3$ and $x=2$

Then, $y=9$ when $x=-3$

And $y=4$ when $x=2$

  $\left(-3,9\right)$ and $\left(2,4\right)$ are the points where the parabola and line $\left(2\right)$ intersect

From equation $\left(3\right)$ , we have

  $\begin{array}{l}x=y+6\\ y=x-6\end{array}$

Substituting into equation $\left(1\right)$ , we have

  $\begin{array}{l}{x}^2-yc+6\\ Or\\ x^2-x+6=0\end{array}$

  $x=$ imaginary

Then, line $\left(3\right)$ and the parabola do not intersect

  

Conclusion: The parabola do not intersect