Sketch the graph and find the RADIUS and EQUATION of a circle in standard form of the given endpoints . (show all the solution) Diameter with endpoints (-2,-1) and (6,3)

The center (h,k) is the midpoint of the diameter, right? (2,1)

The radius is the distance from the center to either endpoint: r=√(4^2+2^2)
The equation is (x-h)^2 + (y-k)^2 = r^2

not much calculus here -- just Algebra 2

To find the radius and equation of the circle, we need to first determine the center of the circle using the midpoint formula.

The midpoint formula is given by:
Midpoint = ((x1+x2)/2, (y1+y2)/2)

In this case, the coordinates of the endpoints are (-2, -1) and (6, 3).

Using the midpoint formula, we can calculate the coordinates of the center of the circle as follows:

Midpoint = ((-2 + 6)/2, (-1 + 3)/2)
= (4/2, 2/2)
= (2, 1)

Therefore, the center of the circle is (2, 1).

The radius of the circle is equal to half the distance between the two endpoints. Using the distance formula:

Distance = √((x2-x1)^2 + (y2-y1)^2)

In this case, using the endpoints (-2, -1) and (6, 3):

Distance = √((6 - (-2))^2 + (3 - (-1))^2)
= √(8^2 + 4^2)
= √(64 + 16)
= √80
= 4√5

Therefore, the radius of the circle is 4√5.

Now, we can write the equation of the circle in standard form, which is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) represents the coordinates of the center and r represents the radius.

Substituting the values we found, we get:

(x - 2)^2 + (y - 1)^2 = (4√5)^2
(x - 2)^2 + (y - 1)^2 = 80

So, the equation of the circle in standard form is (x - 2)^2 + (y - 1)^2 = 80.

To sketch the graph of a circle with the given endpoints as the diameter, we can follow these steps:

1. Find the center point of the circle by taking the average of the x-coordinates and the y-coordinates of the two given endpoints.

Center point (h, k) = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Given endpoints: (-2, -1) and (6, 3)
Center point = ((-2 + 6)/2, (-1 + 3)/2) = (2, 1)

So, the center of the circle is (2, 1).

2. Find the radius of the circle by using the distance formula between the center point and one of the endpoints.

The distance formula is given by:

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

Using one of the endpoints (-2, -1):

Distance = √((6 - (-2))² + (3 - (-1))²) = √(8² + 4²) = √(64 + 16) = √80 = 4√5

So, the radius of the circle is 4√5.

3. Write the equation of the circle in standard form using the center point (h, k) and radius r.

The equation of the circle in standard form is:

(x - h)² + (y - k)² = r²

Center point (h, k) = (2, 1)
Radius r = 4√5

(x - 2)² + (y - 1)² = (4√5)²
(x - 2)² + (y - 1)² = 80

So, the graph of the circle has a center at (2, 1) and a radius of 4√5. The equation of the circle in standard form is (x - 2)² + (y - 1)² = 80.