Question

Help please!!! :')

Given a diameter of a circle at endpoints (8,-7) and (4,5), algebraically find the equation of the circle.

thank you smm <3

Answer 1

Answer:
`(x - 6)^2 + (y + 1)^2 = 40`

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Step-by-step explanation:

The x coordinates of the given endpoints are x = 8 and x = 4. Apply the midpoint to them to get
`(x_1+x_2)/(2) = (8+4)/(2) = (12)/(2) = 6` which is the x coordinate of the midpoint. I added them up and divided by 2.

Similar steps will have the y coordinate of the midpoint be

`(y_1+y_2)/(2) = (-7+5)/(2) = (-2)/(2) = -1`

Overall, the midpoint of those given points is (6, -1)

This is the center of the circle because the two endpoints are part of a diameter.

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Now turn to the template

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

We will plug in the center (h,k) = (6, -1) and one of the endpoints for (x,y). Let's say we go for (x,y) = (4,5)

This leads us to the following:

`(x - h)^2 + (y -k )^2 = r^2\\\\(4 - 6)^2 + (5 - (-1) )^2 = r^2\\\\(4 - 6)^2 + (5 + 1 )^2 = r^2\\\\(-2)^2 + (6 )^2 = r^2\\\\4 + 36 = r^2\\\\r^2 = 40\\\\`

We don't need to isolate r itself. If you wanted, you could take the square root of both sides to see that the radius is
`r = √(40) \approx 6.3246`. However, again we don't need to solve for r entirely when forming the circle's equation.

With that
`r^2` value in mind, and the h & k values as well, we go from this template

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

to this final answer

`(x - 6)^2 + (y + 1)^2 = 40`

Answer 2

(x-6)^2 + (y+1)^2=40

Step-by-step explanation:

see image. Use Midpoint Formula to find the center. Then use Distance Formula to find the diameter. Cut the diameter in half to find the radius. Use the circle formula to insert (h,k) the center and r, the radius to find the equation. See image.