Answer:
The standard form of the equation of a circle with center at (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
We are given that the center of the circle is (4, -1), so h = 4 and k = -1. We also know that the circle passes through the point (-4, 1), which means that the distance from the center of the circle to (-4, 1) is the radius of the circle.
The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
So the radius of the circle is:
r = sqrt((-4 - 4)^2 + (1 - (-1))^2) = sqrt(100) = 10
Now we can substitute the values of h, k, and r into the standard form equation of a circle:
(x - 4)^2 + (y + 1)^2 = 10^2
Expanding the equation gives:
x^2 - 8x + 16 + y^2 + 2y + 1 = 100
Simplifying and putting the equation in standard form, we get:
x^2 + y^2 - 8x + 2y - 83 = 0
Therefore, the equation in standard form of the circle with center at (4, −1) and that passes through the point (−4, 1) is:
x^2 + y^2 - 8x + 2y - 83 = 0