What is the general form of the equation for the given circle? A (4, 7) B (7,4)1.x2 + y2 − 8x − 8y + 23 = 02.x2 + y2 − 8x − 8y + 32 = 03.x2 + y2 − 4x − 4y + 23 = 04.x2 + y2 + 4x + 4y + 9 = 0

Answer: the first option: x^2 + y^2  -8x -8y + 23 = 0

Explanation:

The equation of a circle is (x - xo)^2 + (y - yo)^2 = r^2, where (xo,yo) is the center of the circle and r is the radius of the circle.

From the figure you can calculate both the center (xo,yo) and the radius (r), using a fundamental assumption: that the two segments drawn form a 90° angle.

1) Center:

Note that the y-coordinate of the point B(7,4) is the same y-coordinate of the center of the circle, so yo = 4

And the x-coordinate of the point A(4,7) is the same x-coordinate of the center of the circle, so xo = 4

Then, the center is (4,4).

2) Radius, r

The radius may be calculated as the difference between the x-coordinates of the two points, i.e. 7 - 4 = 3.

Of course, it may also be calculated as the difference of the y-coordinates of the same two points, i.e.: 7 - 4 = 3.

Then, r = 3.

3) use the formula (x - xo)^2 + (y - yo)^2 = r^2

(x - 4)^2 + (y - 4)^2 = 3^2

And expand the parenthesis:

x^2 - 8x + 16 + y^2 - 8y + 16 = 9

x^2 - 8x + y^2 - 8y + 23 = 0

That is the first option: x^2 + y^2  -8x -8y + 23 = 0