The equation of the circle in standard form is; (x - 7)² + (x + 6)² = 26
The steps that can be used to find the equation of the circle can be presented as follows;
The radius from the center of the circle to the tangent is perpendicular to the tangent, therefore;
The equation of the radius of the circle can be found as follows;
Slope of the radius of the circle = 1/5
Equation of the radius is; (y - (-7)) = (1/5) × (x - 2)
y + 7 = (1/5) × (x - 2)
5·y + 35 = x - 2
x = 5·y + 35 + 2
x = 5·y + 37
The coordinates of the center is therefore, the location of the intersection of the radius and the line x = 2·y + 19, which can be found as follows;
Equating the equation of the radius to the equation of the line passing through the center, we get;
5·y + 37 = 2·y + 19
5·y - 2·y = 19 - 37
3·y = -18
y = -18/3
y = -6
x = 5·y + 37
x = 5 × (-6) + 37
x = 7
The coordinates of the center of the circle is; (7, -6)
The length of the radius of the circle is therefore;
r = √((2 - 7)² + (-7 - (-6))²)
r = √(26)
The equation of the circle is therefore;
(x - 7)² + (y - (-6))² = 26
(x - 7)² + (y + 6)² = 26
The complete question, obtained from a similar question found through can be presented as follows;
The circle is tangent to the line y = 3 - 5·x at the point (2, -7) whose center is on the line x = 2·y + 19 find the equation of the circle