The radius of a circle given C (4,-1) and (8,2). What is the equation that defines the circle if its center is at the fourth quadrant?

if you mean that the center is (4,-1) and (8,2) lies on the circle, then

r^2 = (8-4)^2+(2+1)^2 = 25

so the circle is

(x-4)^2 + (y+1)^2 = 25

Math

To find the radius and equation of a circle with a given center and two points on the circle, follow these steps:

Step 1: Find the center of the circle.
Given that the center of the circle is in the fourth quadrant, we know that the x-coordinate will be positive, and the y-coordinate will be negative.

Let's find the center using the given points (4, -1) and (8, 2):
x-coordinate of center = (x-coordinate of point 1 + x-coordinate of point 2) / 2
= (4 + 8) / 2
= 12 / 2
= 6

y-coordinate of center = (y-coordinate of point 1 + y-coordinate of point 2) / 2
= (-1 + 2) / 2
= 1 / 2
= 0.5

So, the center of the circle is (6, 0.5).

Step 2: Find the radius of the circle.
The radius is the distance between the center of the circle and any point on the circle. We can use the distance formula to find the radius.

Distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Let's find the distance between the center (6, 0.5) and one of the given points, say (4, -1):
Distance = √((4 - 6)^2 + (-1 - 0.5)^2)
= √((-2)^2 + (-1.5)^2)
= √(4 + 2.25)
= √6.25
= 2.5

So, the radius of the circle is 2.5.

Step 3: Write the equation of the circle.
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2

Using the values we found, the equation of the circle is:
(x - 6)^2 + (y - 0.5)^2 = 2.5^2

Simplifying:
(x - 6)^2 + (y - 0.5)^2 = 6.25

The equation that defines the circle with a center in the fourth quadrant and passing through the points (4, -1) and (8, 2) is:
(x - 6)^2 + (y - 0.5)^2 = 6.25

To find the equation of a circle given its center and a point on the circle, we will use the formula:

(x - h)² + (y - k)² = r²

Where:
- (h, k) is the center of the circle
- r is the radius of the circle

In this case, you mentioned that the center of the circle is in the fourth quadrant. Since the fourth quadrant has positive x- and y-coordinates, the center of the circle will have positive values.

First, let's find the center of the circle using the given points. The center is the midpoint between the two points (C1 and C2):

Center:
- x-coordinate of the center = (x-coordinate of C1 + x-coordinate of C2) / 2
= (4 + 8) / 2
= 12 / 2
= 6
- y-coordinate of the center = (y-coordinate of C1 + y-coordinate of C2) / 2
= (-1 + 2) / 2
= 1 / 2
= 0.5

So, the center of the circle is (6, 0.5).

Next, we need to find the radius of the circle, which is the distance between the center and one of the given points. Let's use the distance formula:

Distance between two points (x₁, y₁) and (x₂, y₂):
- √((x₂ - x₁)² + (y₂ - y₁)²)

Let's use the center (6, 0.5) and one of the given points (C1 - 4, -1) to find the radius:

Radius:
- x₁ = 6
- y₁ = 0.5
- x₂ = 4
- y₂ = -1

r = √((4 - 6)² + (-1 - 0.5)²)
= √((-2)² + (-1.5)²)
= √(4 + 2.25)
= √6.25
= 2.5

So, the radius of the circle is 2.5 units.

Now we have all the information needed to write the equation of the circle:

(x - 6)² + (y - 0.5)² = (2.5)²

Simplifying it, we get:

(x - 6)² + (y - 0.5)² = 6.25

Therefore, the equation that defines the circle with its center at the fourth quadrant is (x - 6)² + (y - 0.5)² = 6.25.