If we denote the center of the circle by the point (h = −1 , k = 2) then the circle eaution is:
By implicit differentiation of the circle equation, we will find the slope of any tangent line to the circle.
Once we know the radius and the center of the circle, we have the equation of the circle as:
After multipling the paranthesis we get the equation of the line as:
x2 + y2 − 4x + 6y + 4 = 0
In order to find the intersection points with the y axis substitute x = 0 in the circle's equation.
We get the quadratic equation: y2 + 6y + 4 = 0
And the intersection points are: (0 , −0.76) and (0 , −5.24)
The perpendicular line equation is: y − yc = M (x − xc)
4x − 3y + 1 = 0
Solving this line equation and the given line equation to get the intersection point which is also located on the circle.
We get the solutions x = −0.4 and y = −0.2
And the radiuse is equal to 5.
Hence the equation of the circle is:
(x − 5)2 + (y + 5)2 = 52
If we denote the equation of a line as:
y = ax + b
The value of h is the shift value along the x axis when y = 0 and is equal to h = −b / a
The equations of the upper right and lower left circles (blue circles) are:
The equations of the lower left and upper right circles (green circles) are: Notice that the angle now is (π − α)/2)
Notice that each circle center is located at a different quadrant.