Chapter 10.9, Problem 34E

To determine

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

Answer to Problem 34E

The circle and the line do not intersect in the real axes

Explanation of Solution

Given:

  $\{\begin{array}{l}x>0\\ y>0\\ x+y<10\\ x^2+y^2>9\end{array}$

Calculation:

The given function

  $\{\begin{array}{l}x>0\\ y>0\\ x+y<10\\ x^2+y^2>9\end{array}$

Here, it is given that $x+y<10$ and $x^2+y^2>9$

Then we take

  $\begin{array}{l}x+y=\mathrm{10..........}(1)\\ \frac{x}{10}+\frac{y}{10}=1\end{array}$

Then the line passes through points $(10,0)$ and $(0,10)$

And $x^2+y^2=\mathrm{9.........}(2)$

Then the center of the circle is $(0,0)$ and the radius is $3$

  $(3,0),(-3,0)$ , and $(0,-3)$ are points on the circle

From equation $(1)$ , we have

  $\begin{array}{l}x+y=10\\ y=10-x\end{array}$

Substituting this expression into equation $(2)$ , we get

  $\begin{array}{l}{x}^2+{(}^{10}=9\\ x^2+100+x^2-20x=9\\ 2x^2-20x+91=0\\ x^2-10x+\frac{91}{2}=0\end{array}$

  $x$ is imaginary so y is also imaginary

Therefore, the circle and the line do not intersect in the real axes

  

Conclusion:

Therefore, the circle and the line do not intersect in the real axes