x

can be reduced to the location of the intersection points of the two conics

y = x

yx + ay + bx + c =0.

The first is always the same well known parabola. The second is a rectangular hyperbola discussed in detail in Rectangular_Hype_From_Line.html .

Its center is at (-a, -b) and its asymptotes are parallel to the axes. The coefficients (a,b) are in reverse order from those in the above reference.

The figure below illustrates the method:

1) First draw the line bx+ay+c=0 and find its intersection points {A,B} with the axes.

2) Then find their symmetrics {A',B'} with respect to (-a,-b).

3) Find the orthocenter H of one of the triangles formed by the parallelogram ABB'A' and its diagonals (here BAB').

4) Pass a conic throught the five points {A,B,A',B',H}. This coincides with yx+ay+bx+c=0.

5) Draw the parabola y=x

6) Find the intersection point(s) (s

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