help please!!!!!!!!!!!!!!!!!!!!!!!!!! really need it!!?

Question

write the equation of the circle with the given conditions:
1) center (4,9) goes through (7,12)
2) diameter endpoints are (4,-8) and (-2,6)

please help!!!thanks

Answer 1

PROBLEM 1:

The general formula for a circle is:
(x - x1)² + (y - y1)² = r²

So far you know that the center is (4, 9), so that would be:
(x - 4)² + (y - 9)² = r²

Okay, but you aren't told what r² is, but you are told another point. So plug in the other point (7, 12) to find out what r² is:

(7 - 4)² + (12 - 9)² = r²
3² + 3² = r²
9 + 9 = r²
18 = r²

So the final equation for the circle is:
(x - 4)² + (y - 9)² = 18

PROBLEM 2:

Okay, here the method is the same, except you are told the diameter endpoints. That means the center of the circle will be the midpoint.

The midpoint of two points will just be the halfway point. Take average of the x-coordinates and the average of the y-coordinates to get the midpoint:

midpoint (4, -8) and (-2, 6)
x-coodinate = (x2 + x1)/2
= (-2 + 4) / 2
= 2 / 2
= 1

y-coordinate = (y2 + y1)/2
= (6 + (-8)) / 2
= (-2)/2
= -1

So the midpoint (center) will be at (1, -1)

Now use the method above.

Plug in the center point (1, -1):
(x - 1)² + (y - (-1))² = r²

Simplify:
(x - 1)² + (y + 1)² = r²

Again you need to figure out r², so plug in one of your points... I'll pick (-2, 6):
(-2 - 1)² + (6 + 1)² = r²
(-3)² + 7² = r²
9 + 49 = r²
58 = r²

So the final formula for the circle will be:
(x - 1)² + (y + 1)² = 58