Write the standard form

Question

Write the standard form of the equation of the circle with the given characteristics.

Endpoints of a diameter: (0, 0), (8, 6)

Answer 1

Hi Nikki G

Since you are given the endpoints of the diameter (0,0) and (8,6), and the midpoint of the diameter is also the center of the circle

I have an alternate route toward (x - h)² + (y - k)² = r²

First start by calculating the midpoint of the diameter, this gives you the h,k coordinates, using only what you are given

Midpoint formula is (x₁ + x₂)/2 for the x coordinate and , (y₁+y₂)/2 for the y coordinate

For the center of a circle

x = h = (0+8)/2 = 4

y = k = (0+6)/2 = 3

Secondly you now just plug these values into (x - h)² + (y - k)² = r²

Finally you already have two points along the clrcle either pair can be used to find r then another can be used to check it

Your given points (0, 0), (8, 6)

(0 - 4)² + (0 - 3)² = r²

(-4)² + (-3)² = r²

16 + 9 = r²

25 = r²

√25 = r

5 = r

Checking the radius with other given coordinates

(8 - 4)² + (6 - 3)² = r²

(4)² + (3)² = r²

16 + 9 = r²

25 = r²

You still get your equation as

(x - 4)² + (y - k)² = 25

You can graph your circle to confirm the given points along the circlen the radius and the center.

Answer 2

Given the standard form of a circle: , where h and k are your x and y coordinates for the center of the circle and r is the radius, we need to find the radius and center of the circle from the information given to complete the equation.

The radius of the circle can be found using the distance formula, . Alternatively, you can use the pythagorean theorem.

Plugging in the two points, the distance is 10, so our radius is 5. Now to find the center, we simply need to find the midpoint between the two points. In this case, this is (4,3).

Now we can plug this information into the standard form equation, and we will obtain our final answer: .

Please comment if you have any questions!