Pre-AP PreCalculus

A BRIEF REVIEW OF THE CONIC SECTIONS

THE TECHNICAL DEFINITIONS:

Circle

The collection of points in the Cartesian Plane which are equidistant from a fixed point known as the center.

In the diagram below, the center is the point C(h,k) and the fixed distance is labeled as r, the radius of the circle.

Parabola

The collection of points in the Cartesian Plane which are equidistant from a fixed point called the focus and the a fixed line known as the directrix.

In the diagrams below, the first parabola has the y-axis as its axis of symmetry, while the second parabola has the x-axis as its axis of symmetry.

Ellipse

The collection of points in the Cartesian Plane such that the sum of the distances from two fixed points called the foci is constant.

In the diagrams below, the first ellipse has its major axis on the x-axis and its minor axis on the y-axis, while the second ellipse has its major axis on the y-axis and its minor axis on the x-axis. The foci are located at the points (c, 0), (-c,0) and (0, c), (0, -c) respectively.

Hyperbola

The collection of points in the Cartesian Plane such that the absolute value of the differences of the distances from two fixed points known as the foci is constant.

In the diagrams below, the first hyperbola has its transverse axis on the x-axis and its conjugate axis on the y-axis, while the second hyperbola has its transverse axis on the y-axis and its conjugate axis on the x-axis. The foci are located at the points (a, 0), (-a,0) and (0, a), (0, -a) respectively.

HISTORY OF THE CONIC SECTIONS

The words circle, ellipse, hyperbola, and parabola were first used by the members of the Pythagorean society in ancient Greece around 540 B.C. These terms were used in connection with the regions between the curves instead of the curves themselves as we refer to them today. Menaechmus (350 B.C.) is credited with the first treatment of the terms as curves generated by sections of geometric solids. However, it was Appolonius of Perga (about 225 B.C.), an astronomer of some fame in Greece, who wrote an eight-book essay entitled Conic Sections . His work difered from others in that he obtained all of the conic sections from one double right cone intersected by a plane.

Appolonius stated that all variations in the shape of a conic section could be obtained by varying the slope of the plane intersecting the conical surface.

Before you scroll down any further, take a moment or two and prove to yourself that Appolonius' claim was indeed correct!

THE ANSWERS

The Circle	The Ellipse	The Parabola	The Hyperbola

SOME THINGS TO THINK ABOUT

In PreCalculus we are going to continue our study of the conic sections by considering the solutions to systems of conics contained in the Cartesian Plane, that is, multiple conic sections and how they can intersect each other in the x-y plane.

For example, two circles can intersect in each of the following ways:

• 0 points - two non-overlapping circles
  
• 1 point - two circles which are externally tangent
  
• 2 points - two overlapping circles
  
• infinitely many points - two identical circles laying upon one another

TONIGHT'S HOMEWORK ASSIGNMENT

For homework tonight, consider 7 of the following 9 cases involving systems of conic sections. For each pair, you must consider the different ways that they can intersect one another, list the different possibilities for the number of intersection points, and include a diagram which supports your claim.

1. A circle and a parabola
   
2. A circle and an ellipse
   
3. A circle and a hyperbola
   
4. Two parabolas
   
5. A parabola and an ellipse
   
6. A parabola and a hyperbola
   
7. Two ellipses
   
8. An ellipse and a hyperbola
   
9. Two hyperbolas
   

FOR NEXT CLASS

For the next class, you must review and have memorized the following formulas (these are all in your Algebra II notes from last year):