Answer:
Radius: 

Step-by-step explanation:
Given

Solving (a): The radius of the circle
First, we express the equation as:

Where


So, we have:

Divide through by 9

Rewrite as:

Group the expression into 2
![[x^2 + 3x] + [y^2+ \frac{12}{9}y] =- \frac{19}{9}](https://tex.z-dn.net/?f=%5Bx%5E2%20%20%2B%203x%5D%20%2B%20%5By%5E2%2B%20%5Cfrac%7B12%7D%7B9%7Dy%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D)
![[x^2 + 3x] + [y^2+ \frac{4}{3}y] =- \frac{19}{9}](https://tex.z-dn.net/?f=%5Bx%5E2%20%20%2B%203x%5D%20%2B%20%5By%5E2%2B%20%5Cfrac%7B4%7D%7B3%7Dy%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D)
Next, we complete the square on each group.
For ![[x^2 + 3x]](https://tex.z-dn.net/?f=%5Bx%5E2%20%20%2B%203x%5D)
1: Divide the 
2: Take the 
3: Add this 
So, we have:
![[x^2 + 3x] + [y^2+ \frac{4}{3}y] =- \frac{19}{9}](https://tex.z-dn.net/?f=%5Bx%5E2%20%20%2B%203x%5D%20%2B%20%5By%5E2%2B%20%5Cfrac%7B4%7D%7B3%7Dy%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D)
![[x^2 + 3x + (\frac{3}{2})^2] + [y^2+ \frac{4}{3}y] =- \frac{19}{9}+ (\frac{3}{2})^2](https://tex.z-dn.net/?f=%5Bx%5E2%20%20%2B%203x%20%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E2%5D%20%2B%20%5By%5E2%2B%20%5Cfrac%7B4%7D%7B3%7Dy%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E2)
Factorize
![[x + \frac{3}{2}]^2+ [y^2+ \frac{4}{3}y] =- \frac{19}{9}+ (\frac{3}{2})^2](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%5E2%2B%20%5Cfrac%7B4%7D%7B3%7Dy%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E2)
Apply the same to y
![[x + \frac{3}{2}]^2+ [y^2+ \frac{4}{3}y +(\frac{4}{6})^2 ] =- \frac{19}{9}+ (\frac{3}{2})^2 +(\frac{4}{6})^2](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%5E2%2B%20%5Cfrac%7B4%7D%7B3%7Dy%20%2B%28%5Cfrac%7B4%7D%7B6%7D%29%5E2%20%5D%20%3D-%20%5Cfrac%7B19%7D%7B9%7D%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E2%20%2B%28%5Cfrac%7B4%7D%7B6%7D%29%5E2)
![[x + \frac{3}{2}]^2+ [y +\frac{4}{6}]^2 =- \frac{19}{9}+ (\frac{3}{2})^2 +(\frac{4}{6})^2](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B4%7D%7B6%7D%5D%5E2%20%3D-%20%5Cfrac%7B19%7D%7B9%7D%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E2%20%2B%28%5Cfrac%7B4%7D%7B6%7D%29%5E2)
![[x + \frac{3}{2}]^2+ [y +\frac{4}{6}]^2 =- \frac{19}{9}+ \frac{9}{4} +\frac{16}{36}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B4%7D%7B6%7D%5D%5E2%20%3D-%20%5Cfrac%7B19%7D%7B9%7D%2B%20%5Cfrac%7B9%7D%7B4%7D%20%2B%5Cfrac%7B16%7D%7B36%7D)
Add the fractions
![[x + \frac{3}{2}]^2+ [y +\frac{4}{6}]^2 =\frac{-19 * 4 + 9 * 9 + 16 * 1}{36}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B4%7D%7B6%7D%5D%5E2%20%3D%5Cfrac%7B-19%20%2A%204%20%2B%209%20%2A%209%20%2B%2016%20%2A%201%7D%7B36%7D)
![[x + \frac{3}{2}]^2+ [y +\frac{4}{6}]^2 =\frac{21}{36}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B4%7D%7B6%7D%5D%5E2%20%3D%5Cfrac%7B21%7D%7B36%7D)
![[x + \frac{3}{2}]^2+ [y +\frac{4}{6}]^2 =\frac{7}{12}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B4%7D%7B6%7D%5D%5E2%20%3D%5Cfrac%7B7%7D%7B12%7D)
![[x + \frac{3}{2}]^2+ [y +\frac{2}{3}]^2 =\frac{7}{12}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B2%7D%7B3%7D%5D%5E2%20%3D%5Cfrac%7B7%7D%7B12%7D)
Recall that:

By comparison:

Take square roots of both sides

Split

Rationalize





Solving (b): The center
Recall that:

Where


From:
![[x + \frac{3}{2}]^2+ [y +\frac{2}{3}]^2 =\frac{7}{12}](https://tex.z-dn.net/?f=%5Bx%20%2B%20%5Cfrac%7B3%7D%7B2%7D%5D%5E2%2B%20%5By%20%2B%5Cfrac%7B2%7D%7B3%7D%5D%5E2%20%3D%5Cfrac%7B7%7D%7B12%7D)
and 
Solve for h and k
and 
Hence, the center is:

Production possibilities frontier, PPF, is defined as a representation of the point at which a country’s economy is most efficiently producing its goods and services. Efficient production means allocating its resources in the best way possible.
<span>
The points in the PPF shifts outward. The more efficient production is the more the PPF shifts outward. With this in mind, any point found inside the PPF means that there is inefficient resource allocation. If the PPF shifts inward, it means that the efficiency production is regressing and available resources are not fully used to its potential which will lead to a downward turn of the economy.
</span>
From the second:
2x-y=4
x=(4+y)/2 and applying this to first:
4+y+3y=12
4y+4=12
4y=8
y=2 and since x=(4+y)/2, x=3 so
x+y=2+3=5
Each time they assume the sum<span> is </span>rational<span>; however, upon rearranging the terms of their equation, they get a contradiction (that an </span>irrational number<span> is equal to a </span>rational number<span>). Since the assumption that the </span>sum of a rational<span> and </span>irrational number<span> is </span>rational<span>leads to a contradiction, the </span>sum<span> must be </span>irrational<span>.</span>