Find the standard form for the equation of a circle

Question

(x − h) ^2 + (y − k) ^2 = r^2

with a diameter that has endpoints (− 10, 7) and (10, − 6).

Answer 1

Hi, Bailey!

As you've given, the standard equation for a circle is of the form (x - h)² + (y - k)² = r², where r represents the radius of the circle and the coordinate pair (h, k) is the center of the circle. Therefore, we need to use the two given points to solve for the radius and center of the circle in order to write out our full equation.

Let's start by first finding the center of the circle. We are given coordinate pairs that represent the endpoints of the diameter of the circle. Of note, the diameter of a circle passes directly through the center of a circle; specifically, the midpoint of the diameter is located at the center of the circle. Therefore, we can use the midpoint formula to find the center of the circle.

Midpoint = ( [x₁ + x₂] / 2 , [y₁ + y₂] / 2 ), where (x₁, y₁) and (x₂, y₂) are the endpoints.

Let (x₁, y₁) = (-10, 7)

Let (x₂, y₂) = (10, -6)

Midpoint = ( [-10 + 10] / 2 , [7 + -6] / 2 )

Midpoint = (0, 0.5)

Therefore, the center of our circle is (0, 0.5) = (h, k)

Next, we need to solve for the radius of the circle. Since we already know the center, we can rewrite our general formula as follows:

(x - 0)² + (y - 0.5)² = r²

x² + (y - 0.5)² = r²

Next, we can choose either one of the points that we are given on the circle, and plug them into our x and y coordinates, which will allow us to solve for r². Let's choose (-10, 7):

(-10)² + (7 - 0.5)² = r²

(-10)² + (6.5)² = r²

r² = 142.25

Now, of note, this value of r² represents the radius squared, not the actual radius itself. In order to get the radius, you would have to take the square root of this number. However, since the question is only asking for the equation, we only need r² anyway.

We can therefore finally plug in our values of (h, k) and r² into our general formula to get our final answer:

x² + (y - 0.5)² = 142.25

I hope that this response was helpful, and please don't hesitate to reach out if you have any further questions!

Answer 2

diameter = sqr(10--10)^2^2 +(7- -6)^2) =sqr569

radius = sqr569/2

center = midpoint = average of the two endpoints = (0, 1/2)

x^2 + (y-1/2)^2 = (sqr569/2)^2 is standard form

(x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r=radius = half the diameter

h=0, k=1/2, r = (sqr569)/2