Conic sections FAQ

Conic sections are the shapes you get when you slice a cone at different angles. There are four types of conic sections: ellipses, hyperbolas, parabolas, and circles.

Conic sections show up in a lot of places! For example, the orbits of planets around the sun are elliptical. Hyperbolas are often used in the design of telescopes and antennas. Parabolas are important in physics, as they describe the shape of projectiles in flight.

The standard equation of a circle looks like $(x-h{)}^2+(y-k{)}^2=r^2$‍. When the equation is in this form, the center of the circle is located at $(h,k)$‍ and has a radius of $r$‍.

For example, the equation $(x+2{)}^2+(y-3{)}^2=16$‍ represents a circle with center $(-2,3)$‍ and radius $4$‍.

A parabola can be graphed as the set of all points that are equidistant from a fixed point (the "focus") and a fixed line (the "directrix").

We use the fact that any point $(x,y)$‍ on the parabola must be equidistant from the focus and directrix.

Consider, for example, the parabola whose focus is at $(-3,6)$‍ and directrix is $y=4$‍. We start by assuming a general point on the parabola $(x,y)$‍.

Using the distance formula, we find that the distance between $(x,y)$‍ and the focus $(-3,6)$‍ is $\sqrt{(x+3{)}^2+(y-6{)}^2}$‍, and the distance between $(x,y)$‍ and the directrix $y=4$‍ is $\sqrt{(y-4{)}^2}$‍. On the parabola, these distances are equal: