Write the equation for a circle with a radius of 5 passing through points A (-4: 0) B (4: 2).

The canonical equation of a circle on the coordinate plane Oxy with center at point C (xc; yc) and radius R has the form (x – xc) ^ 2 + (y – yc) ^ 2 = R ^ 2.
By the condition of the assignment R = 5. The coordinates xс and yс of the center С of the given circle are unknown. If the indicated circle passes through points A (–4; 0) and B (4; 2) (that is, points A and B belong to a given circle), then the coordinates of these points must satisfy the equation of the circle, that is, when substituting the coordinates of points A (- 4; 0) and B (4; 2) in the equation of the circle, must obtain correct equalities.
Therefore, we obtain the following two equations for the unknown coordinates xc and yc of the center C of the given circle: ((–4) – xc) ^ 2 + (0 – yc) ^ 2 = 5 ^ 2 and (4 – xc) ^ 2 + (2 – mustache) ^ 2 = 5 ^ 2. These equations allow us to establish: yс = 1 – 4 * xс. Substituting this expression instead of yc in any (for example, in the first) equation, we get 16 + 8 * xc + (xc) ^ 2 + 1 – 8 * xc + 16 * (xc) ^ 2 = 25 or 17 * (xc) ^ 2 = 8, whence xc = ± √ (8/17). Therefore, for xc = √ (8/17), we get yc = 1 – 4 * √ (8/17), similarly, for xc = –√ (8/17), we have yc = 1 + 4 * √ (8/17) …
Thus, we got two equations of the circle: a) (x – √ (8/17)) ^ 2 + (y – 1 + 4 * √ (8/17)) ^ 2 = 25 and b) (x + √ (8 / 17)) ^ 2 + (y – 1 – 4 * √ (8/17)) ^ 2 = 25.
Answers: a) (x – √ (8/17)) ^ 2 + (y – 1 + 4 * √ (8/17)) ^ 2 = 25 and b) (x + √ (8/17)) ^ 2 + (y – 1 – 4 * √ (8/17)) ^ 2 = 25.