Circle and Line
Mutual position of two circles

Mutual position of two circles
Two circles k1 and k2 intersect if the distance between their centers is less than the sum, but greater than
The coordinates of the intersection points of two circles we calculate by solving their equations as system of two quadratic equations,
k1 ::   (x - p1)2 + (y - q1)2 - r1 = 0  and  k2 ::   (x - p2)2 + (y - q2)2 - r22 = 0.
Subtracting the second equation from the first gives equation of the line k1- k2 = 0 through the intersection
points A and B of the circles.
Coordinates of intersections A and B satisfy equations of the circles k1 and k2 and the equation of the line  k1- k2 = 0.
This line is called the radical line and represents the locus or set of all points in the plane of equal power with respect to two nonconcentric circles.
The radical line is a line perpendicular to the line connecting the centers of the two circles.
 Since the slope of the line S1S2,
 then is the slope of the radical line.
If two circles touch each other outside then, the radical line is at the same time their common tangent.
The pole and the polar
Given is a circle k ::  x2 + y2  = rand a point P(x0, y0) outside the circle. The contact points of tangents from P to the circle, are at the same time the intersections of the given circle k and the circle k' whose center is the midpoint of the line segment OP, that is
Therefore, the equation of the circle k' is

or, after squaring and reducing
 k' :: x2 + y2 - x0 x - y0 y = 0.
Equation of the line through tangency points, which is perpendicular to the line OP, is
 p ::   k -  k' = 0  or x0 x + y0 y = r2.
This line is called the polar of the point P with respect to the circle, and point P is called the pole of the polar.
If given is a translated circle (x - p)2 + (y - q)2 = r2 with the center at the point S(p, q), then the equation
of the polar of the point
P(x0, y0) is,
 (x0 - p) · (x - p) + (y0 - q) · (y - q) = r2.
Angle between two circles
Angle between two circles is defined as the angle between the two tangent lines at any of the intersection
points of the circles.
Example:   Given are circles k1 ::  x2 + y2 - 4x - 6y + 3 = 0 and  k2 ::  x2 + y2 + 6x + 4y - 7 = 0, find their intersections.
Solution:   Subtracting given equations of circles gives equation of the line through their intersection points called the radical line or radical axis,
 k1 ::   x2 + y2 - 4x - 6y + 3 = 0  (1) k2 ::   x2 + y2  + 6x + 4y - 7 = 0  (2) k1- k2 = 0  =>  -10x - 10y + 10 = 0  or  y = -x + 1. plugging  y = -x  + 1 into  (1) x2 + (-x + 1)2 - 4x - 6(-x + 1) + 3 = 0 or  2x2 - 2 = 0,  so  x1 = -1 and  x2 = 1, y = -x + 1  =>    y1 = 2  and  y2 = 0. thus,  A(-1, 2) and B(1, 0).
Example:   From the point A(2, -2) drawn are tangents to the circle (x + 3)2 + (y -1)2 = 17, find equations of tangents using the polar and taking the point A as the pole.
Solution:   Coordinates of the point A plug into equation of the polar,
 so,  P(x0, y0)  or  A(2, -2),   S(-3, 1)  and  r2  = 17 (x0 - p) · (x - p) + (y0 - q) · (y - q) = r2 (2 + 3) · (x + 3) + (-2 - 1) · (y - 1) = 17, which gives, p ::  5x - 3y + 1 = 0 - the equation of the polar. By solving system of equations of the polar and the circle we calculate coordinates of points of contact, (1)  5x - 3y + 1 = 0 (2)  (x + 3)2 + (y -1)2 = 17 it follows that,   D1(1, 2) and  D2(-2, -3).
Pre-calculus contents H