Question

the endpoints of the diameter of a circle located at (3,-7) and (5,7). write the equation of the circle​

Answer 1

`(x-4)^2+y^2=50`

Explanation:

Given that the endpoints of the diameter of the circle are (3, -7) and (5, 7), we know that the center of these two points must represent the center of the circle. To determine this, we can use the midpoint formula to find the center,
`C`.

`C = ((x_1 + x_2)/(2), (y_1 + y_2)/(2))`

`C=((3+5)/(2),(-7+7)/(2))`

`C=(4, 0)`

Now, with this in mind, we can figure out the radius,
`r`, of the circle. To do this, we can use the distance formula (or pythagorean theorem) from the center point
`C` to any of the two given endpoints, since we know the distance from either endpoint to the center is equal.

`r=√((C_x-x_1)^2+(C_y-y_1)^2)` (
`C_x` and
`C_y` represent the x and y of the center, while
`x_1` and
`y_1` represent the x and y of either endpoint)

`r=√((4-5)^2+(0-7)^2)`(Using point (5,7))

`r=√((-1)^2+(-7)^2)`

`r=√(1+49)`

`r=√(50)`

Knowing both the center,
`C`, and the radius,
`r`, we can now write a formula for the circle. The formula for a circle is
`(x-h)^2+(y-k)^2=r^2`, where
`(h, k)`represent the center, and
`r` is the radius, as always.

Finally, we get the equation:

`(x-4)^2+(y-0)^2=√(50)^2`

`(x-4)^2+y^2=50`

Below is a graph visualizing the circle and the 2 endpoints.