Write the equation of each circle with radius 5 passing through point (1,8) and its center is on the bisector

Write the equation of each circle with radius 5 passing through point (1,8) and its center is on the bisector of the first coordinate quarter.

(x – x₀) ² + (y – y₀) ² = R² – the kind of equation of the circle, to which we need to come.
(x₀; y₀) – coordinates of the center of the circle;
R – we will have the radius of the circle;
Since the center of the circle lies in the 1st coordinate quarter, therefore x₀> 0, y₀> 0 and x₀ = y₀;
Next, we put the coordinates of the point where our circle passes, as well as the value for the radius of the circle, and take into account that x₀ = y₀, we will compose this equation:
(1 – x₀) ² + (8 – x₀) ² = 5²;
1 – 2x₀ + x₀² + 64 – 16x₀ + x₀² = 25;
2x₀² – 18x₀ + 40 = 0 (divide everything by 2);
x₀² – 9x₀ + 20 = 0;
By Vieta’s theorem:
{x₀₁ * x₀₂ = 20;
{x₀₁ + x₀₂ = 9 => x₀₁ = 4; x₀₂ = 5;
x₀ = y₀ => y₀₁ = 4; y₀₂ = 5;
(4; 4), (5; 5) – centers of circles;

We put the obtained coordinates into the equation of the circle:

(x – 4) ² + (y – 4) ² = 25; (x – 5) ² + (y – 5) ² = 25.