What is required to derive the equations of a parabola, ellipse, and a hyperbola?

What application does Cavalieri's principle have with solid figures?

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.)