Let's solve each part step-by-step to find the center and radius of the circles.
### Part a: [tex]\( x^2 + y^2 = 49 \)[/tex]
The general equation of a circle in standard form is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Here, we have:
[tex]\[ x^2 + y^2 = 49 \][/tex]
Comparing this with the general form, we see:
- The center [tex]\((h, k)\)[/tex] is [tex]\((0, 0)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{49} = 7\)[/tex].
Thus,
- Center: [tex]\((0, 0)\)[/tex]
- Radius: [tex]\(7\)[/tex]
### Part b: [tex]\( (x + 5)^2 + (y - 3)^2 = 9 \)[/tex]
Again, using the standard form:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Here, we have:
[tex]\[ (x + 5)^2 + (y - 3)^2 = 9 \][/tex]
We can rewrite this as:
[tex]\[ (x - (-5))^2 + (y - 3)^2 = 9 \][/tex]
So,
- The center [tex]\((h, k)\)[/tex] is [tex]\((-5, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Thus,
- Center: [tex]\((-5, 3)\)[/tex]
- Radius: [tex]\(3\)[/tex]
### Part c: Endpoints of diameter [tex]\((-1, -1)\)[/tex] and [tex]\((-25, -11)\)[/tex]
To find the center and the radius, we need to:
1. Find the midpoint of the endpoints (this will be the center of the circle).
2. Find the distance between the endpoints and divide by 2 (this will be the radius).
For the midpoint (center):
[tex]\[ \text{Midpoint} = \left( \frac{-1 + (-25)}{2}, \frac{-1 + (-11)}{2} \right) = \left( \frac{-26}{2}, \frac{-12}{2} \right) = (-13, -6) \][/tex]
For the radius:
1. Calculate the distance between the endpoints:
[tex]\[ \text{Distance} = \sqrt{((-25 - (-1))^2 + (-11 - (-1))^2)} = \sqrt{((-25 + 1)^2 + (-11 + 1)^2)} \][/tex]
[tex]\[ = \sqrt{(-24)^2 + (-10)^2} = \sqrt{576 + 100} = \sqrt{676} = 26 \][/tex]
2. Divide the distance by 2 to find the radius:
[tex]\[ \text{Radius} = \frac{26}{2} = 13 \][/tex]
Thus,
- Center: [tex]\((-13, -6)\)[/tex]
- Radius: [tex]\(13\)[/tex]
### Part d: Endpoints of diameter [tex]\((4,0)\)[/tex] and [tex]\((-2,8)\)[/tex]
Similarly to part c, we find the center and the radius.
For the midpoint (center):
[tex]\[ \text{Midpoint} = \left( \frac{4 + (-2)}{2}, \frac{0 + 8}{2} \right) = \left( \frac{2}{2}, \frac{8}{2} \right) = (1, 4) \][/tex]
For the radius:
1. Calculate the distance between the endpoints:
[tex]\[ \text{Distance} = \sqrt{((4 - (-2))^2 + (0 - 8)^2)} = \sqrt{(4 + 2)^2 + (0 - 8)^2} \][/tex]
[tex]\[ = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \][/tex]
2. Divide the distance by 2 to find the radius:
[tex]\[ \text{Radius} = \frac{10}{2} = 5 \][/tex]
Thus,
- Center: [tex]\((1, 4)\)[/tex]
- Radius: [tex]\(5\)[/tex]
In summary:
a. Center: [tex]\((0, 0)\)[/tex], Radius: [tex]\(7\)[/tex]
b. Center: [tex]\((-5, 3)\)[/tex], Radius: [tex]\(3\)[/tex]
c. Center: [tex]\((-13, -6)\)[/tex], Radius: [tex]\(13\)[/tex]
d. Center: [tex]\((1, 4)\)[/tex], Radius: [tex]\(5\)[/tex]
