What is a conic section? A conic section is the result of a plane intersecting various parts of two cones essentially stacked on top of each other (slightly hourglass in shape). Depending upon where the plane intersects the cones, the result will be one of four possibilities: a circle, an ellipse, a parabola, or a hyperbola. All of these shapes have similar characteristics yet have vastly different appearances.
To see an animation of a double cone with the plane z=1, click here.
To see an animation of a double cone with the plane y=2z+1, click here.
To see an animation of a double cone with the plane z=x-2, click here.
To see an animation of a double cone with the plane y=1, click here
So how are these figures constructed? All of these figures are geometrically sound are based upon foci and equidistant line segments.
The circle is the simplest construction since by definition, the circle is the locus of points equidistant from the center.
A parabola is slightly more difficult to construct, but is also based upon the same principle.
We must first construct the directrix line and a focus. Then, we can construct a point on the directrix and draw a line segment from this point to the focus. Once we have the line segment, we can construct the perpendicular bisector. The intersection of the perpendicular bisector with the perpendicular line from the point on the directrix will ultimately be the point to create the parabola.
An ellipse can be constructed from two foci with their sum of the distances being constant as point P travels through the ellipse. We can again construct this using GSP. First, we must construct an arbitrary circle and two foci- one of which is the center of the circle and the other somewhere within the bounds of the interior of the circle. We can construct a line from the center to an arbitrary point on the circle and then construct a line segment from this arbitrary point to the other focus. Their intersection will create an ellipse since these two distances’ sum will be constant.
A hyperbola has a similar construction as the ellipse- the major difference is the presence of one of the foci being located on the exterior of the initial circle. We will again create an arbitrary circle with two foci (one as the center and one lying outside of the circle). Much like the ellipse, we will construct a line from the center to an arbitrary point on the circle and then construct a line segment from this point to the other focus. Their intersection will create a hyperbola.
So how do we go from these geometric figures to complex algebraic equations?
There’s a reason why the textbooks leave this out because we end up with some awful equations!
Let’s first consider the equation of a double cone.
In order to construct a circle, we know we need a plane that is parallel to the x-y plane. So z=d, where d is any real number.
Visually, it is easy to see how these two combined can create the conic section of a circle.
Through simple substitution, we can arrive at the standard equation of a circle given the equation of the double cone.
The derivation of an ellipse is not quite as simple.
We know that we need a plane that comes in at an angle to the z-axis, so the equation should have “slope” for a lack of a better term.
So, we can say that the general equation of the plane to create an ellipse should be y=mz+b. However, due to the complexity of this problem, I will use the equation y=2z+1.
When I attempted this problem for the general case, the problem became a challenge for the completing the square step and I had difficulty getting the problem to become standard form for the ellipse.
Through a similar process, we can see how the parabola can be derived from these basic equations. Again, we need an equation with “tilt” or some kind of “slope”, but through the x-z plane. So we can use z=mx+b as a general equation. Much like the ellipse, these calculations are quite tedious!
Lastly, we can see how a hyperbola can be derived. This time, we need a plane that is perpendicular to the x-y plane, we can use y=d, again where d is any real number.
ß Standard Equation of a hyperbola