Write the equation of a circle that passes through point 8 on the Ox axis and through point 4 on the Oy axis

Write the equation of a circle that passes through point 8 on the Ox axis and through point 4 on the Oy axis, if the center is known to be on the Oy axis.

(x – a) ² + (y – b) ² = R² – the equation of the circle, written in general form, where (a; b) are the coordinates of the center of the circle; R is the radius of the circle. It is known from the problem statement that the equation of the circle passes through point 8 on the Ox axis, that is, through the point with coordinates (8; 0), and through point 4 on the Oy axis, that is, through the point with coordinates (0; 4). In this case, the center is on the Oy axis, which means that the point (0; b) is the center of the circle. Substituting alternately the coordinates of these points into the equation, we obtain a system of two equations with two unknowns:

(8 – 0) ² + (0 – b) ² = R² and (0 – 0) ² + (4 – b) ² = R²;

(8 – 0) ² + (0 – b) ² = (0 – 0) ² + (4 – b) ²;

8² + b² = (4 – b) ²;

b² – 8 ∙ b + 4² – 8² – b² = 0;

8 ∙ b = – 48;

b = – 6, then R = 10, and the equation of the circle will take the form:

x² + (y + 6) ² = 10².

Answer: x² + (y + 6) ² = 10² – the equation of the given circle.