Answer:
You need to have some idea where you want to start if you're going to derive equations for these. You can start with a definition based on focus and directrix, or you can start with a definition based on the geometry of planes and cones. (The second focus is replaced by a directrix in the parabola.) In general, these "conics" represent the intersection between a plane and a cone. Perpendicular to the axis of symmetry, you have a circle. At an angle to the axis of symmetry, but less than parallel to the side of the cone, you have an ellipse. Parallel to the side of the cone, you have a parabola. At an angle between the side of the cone and the axis of the cone, you have a hyperbola. (See source link.)
You can also start with the general form of the quadratic equation.
.. ±((x-h)/a)^2 ± ((y-k)/b)^2 = 1
By selecting signs and values of "a" and "b", you can get any of the equations. (For the parabola, you probably need to take the limit as both k and b approach infinity.)
