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

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Answer 1 · 2017-01-17T06:44:18

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

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

coincident or imaginary.

Let the equations be

$x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 and$ $x^2+y^2+2g_1x+2f_1y+c_1=0 and$

$x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ $x^2+y^2+2g_2x+2f_2y+c_2=0$ .

Subtracting for for common points (x, y)

$2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0$ $2(g_1-g_2)x+2(f_1-f_2)y+c_1-c_2=0$

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

according as

${(f_{1} - f_{2})}_{2}^{2} > = < (g_{1} - g_{2}) (c_{1} - c_{2})$ $(f_1-f_2)^2> =<(g_1-g_2)(c_1-c_2)$ .

Correspondingly, y is real, same or complex.