Algebraic View of the Conic Sections

We now move from a geometric point of view to an algebraic approach. Note that all of the conics that we will see below will satisfy a second-degree equation of the form

where A, B, C, D, E, and F are constants. As we change the values of some of these constants, the shape of the corresponding conic will also change. Hence, it will be very important to focus on these differences in the algebraic equations as we study the individual conic sections. Knowing the subtle differences in the equations will help us to quickly identify the type of conic that is represented by a given equation.

You have undoubtedly already worked with such equations in your previous mathematics courses, although you may not realize it at this point. For example, the equation

is of the form mentioned above. It can be rewritten as

which you may recognize as the equation for a circle centered at the origin with radius 5.

So you see, you (probably) have worked with these algebraic equations before. They will continue to appear as you move into the sections below dealing with the particular conic sections.