Question1. We want to find the equation of the circle with center at (-3, 1) and through the point (2, 13).
Use the distance formula to find the radius.
![r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
![r=\sqrt{(2- - 3)^2+(13-1)^2}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%282-%20-%203%29%5E2%2B%2813-1%29%5E2%7D)
![r=\sqrt{5^2+12^2}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B5%5E2%2B12%5E2%7D)
![r=\sqrt{25+144}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B25%2B144%7D)
![r=\sqrt{169}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B169%7D)
![r = 13](https://tex.z-dn.net/?f=r%20%3D%2013)
The equation of the circle is given by:
![{(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%20%7B%28x%20-%20a%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20b%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
Where (a,b)=(-3,1) is the center and r=13 is the radius.
We substitute to get:
![{(x + 3)}^{2} + ( {y - 1)}^{2} = {13}^{2}](https://tex.z-dn.net/?f=%20%7B%28x%20%2B%203%29%7D%5E%7B2%7D%20%20%2B%20%28%20%7By%20-%201%29%7D%5E%7B2%7D%20%20%3D%20%20%7B13%7D%5E%7B2%7D%20)
![{(x + 3)}^{2} + ( {y - 1)}^{2} = 169](https://tex.z-dn.net/?f=%7B%28x%20%2B%203%29%7D%5E%7B2%7D%20%20%2B%20%28%20%7By%20-%201%29%7D%5E%7B2%7D%20%20%3D%20169)
Question 2) The given circle has equation:
![{x}^{2} + {(y - 6)}^{2} = 50](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%206%29%7D%5E%7B2%7D%20%20%3D%2050)
We want to find the center and radius of this circle.
We need to compare to
![{(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%20a%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20b%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
Rewriting the given equation will make it easy for us;
![{(x - 0)}^{2} + {(y - 6)}^{2} = ( {5 \sqrt{2} )}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%200%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%206%29%7D%5E%7B2%7D%20%20%3D%20%28%20%7B5%20%5Csqrt%7B2%7D%20%29%7D%5E%7B2%7D%20)
Therefore the center is (0,6) and the radius is
![5 \sqrt{2}](https://tex.z-dn.net/?f=5%20%5Csqrt%7B2%7D%20)
Question 3. We want to find the diameter of the circle with equation:
![{(x + 4)}^{2} + ( {y - 9)}^{2} = 18](https://tex.z-dn.net/?f=%7B%28x%20%2B%204%29%7D%5E%7B2%7D%20%20%2B%20%28%20%7By%20-%209%29%7D%5E%7B2%7D%20%20%3D%2018)
By comparing to
![{(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%20a%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20b%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
We have
![{r}^{2} = 18](https://tex.z-dn.net/?f=%20%7Br%7D%5E%7B2%7D%20%20%3D%2018)
![r =3 \sqrt{2}](https://tex.z-dn.net/?f=r%20%3D3%20%20%5Csqrt%7B2%7D%20)
By the diameter is twice the radius.
![diameter = 2 \times 3 \sqrt{2} = 6 \sqrt{2}](https://tex.z-dn.net/?f=diameter%20%3D%202%20%5Ctimes%203%20%5Csqrt%7B2%7D%20%20%3D%206%20%5Csqrt%7B2%7D%20)
Question 4. We want to find the equation of the circle with center at (-3, 0) and diameter 20.
We use the formula:
![{(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%20a%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20b%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
where (a,b)=(-3,0) is the center and r=20 is the radius.
![{(x - - 3)}^{2} + {(y - 0)}^{2} = {20}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%20%20-%203%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%200%29%7D%5E%7B2%7D%20%20%3D%20%20%7B20%7D%5E%7B2%7D%20)
![{(x + 3)}^{2} + {y }^{2} =400](https://tex.z-dn.net/?f=%7B%28x%20%20%2B%203%29%7D%5E%7B2%7D%20%20%2B%20%20%7By%20%7D%5E%7B2%7D%20%20%3D400)
Question 5) We want to find the center and radius of
![{(x - 5)}^{2} + {(y + 2)}^{2} = 16](https://tex.z-dn.net/?f=%7B%28x%20-%205%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20%20%2B%202%29%7D%5E%7B2%7D%20%20%3D%2016)
We compare this equation to
![{(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%7B%28x%20-%20a%29%7D%5E%7B2%7D%20%20%2B%20%20%7B%28y%20%20%20-%20b%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
Then we can see that:
![- a = - 5 \\ a = 5](https://tex.z-dn.net/?f=%20-%20a%20%3D%20%20-%205%20%5C%5C%20a%20%3D%205)
![- b = 2 \\ b = - 2](https://tex.z-dn.net/?f=%20-%20b%20%3D%202%20%5C%5C%20b%20%3D%20%20-%202)
![{r}^{2} = 16](https://tex.z-dn.net/?f=%20%7Br%7D%5E%7B2%7D%20%20%3D%2016)
![r = 4](https://tex.z-dn.net/?f=r%20%3D%204)
Therefore the center is ((5,-2) and the radius is 4.