Question

How to identify equeations of different conics (circle,ellipse,hyperbola)and properties like (centre,foci,etc) given in complex number system(Z).

Answer 1

For an ellipse, there are two foci a,ba,b, and the sum of the distances to both foci is constant. So |z−a|+|z−b|=c|z−a|+|z−b|=c.

For a hyperbola, there are two foci a,ba,b, and the absolute value of the difference of the distances to both foci is constant. So ||z−a|−|z−b||=c||z−a|−|z−b||=c.

For a parabola, there is a focus aa and a line b+ctb+ct (where b,cb,c are complex and the parameter tt is real.) The distances to both must be equal. The distance to the focus is |z−a|