How to write a system of equation that satisfies the conditions "two circles that intersect in three points?"

1 Answer
Jan 17, 2017

Two circles meet in two points that are real and distinct, real and coincident or imaginary.

Explanation:

Two circles meet in two points that are real and distinct, real and

coincident or imaginary.

Let the equations be

#x^2+y^2+2g_1x+2f_1y+c_1=0 and #

#x^2+y^2+2g_2x+2f_2y+c_2=0#.

Subtracting for for common points (x, y)

#2(g_1-g_2)x+2(f_1-f_2)y+c_1-c_2=0#

This quadratic has real and distinct , real and equal or complex

according as

#(f_1-f_2)^2> =<(g_1-g_2)(c_1-c_2)#.

Correspondingly, y is real, same or complex.