Conic Sections in the Plane

Copyright ©1995, 1997, 2001 by James F. Hurley, Department of Mathematics,
University of Connecticut, Unit 3009, Storrs CT 06269-3009. All rights reserved.

     This notebook reviews some basic information about conic sections in the plane.Use it as a memory refresher, or summary of basic facts if you haven't studied conic sections..  

1. Parabolas. A parabola is the set of all points P(x, y) in the plane that are equidistant from a fixed point F (the focus) and a fixed line l (the directrix). If the focus is F(0, p) and the directrix is the line y = -- p, then the parabola has vertex at (0, 0) and equation = 4py. Its graph appears in the following figure. (To generate it, position the cursor after the last line of the code, and press the Enter key on the extreme bottom right of the keyboard.) Note that if p > 0 the curve opens upward, while if p < 0 it opens downward.

                   Plot of Parabola in Standard Position, with vertical axis
    
     Similarly, if the parabola has focus F(p, 0) and directrix x = -- p, then its equation is = 4px. Such a curve opens to the right if p > 0 and to the left if p < 0. You can generate a plot as before: hit the Enter key after you click at the end of the following block of code.   

                  Plot of Parabola in Standard Position, with horizontal axis
        
2. Ellipses. An ellipse is the set of all points P(x, y) the sum of whose distances from two fixed points (the foci) is a constant, 2a. If the foci lie on the x-axis, then the standard equation of such a curve is

+ = 1, where > .
  

Generate a plot by clicking at the end of the following block of code, and hitting the Enter key.

            Plot of Ellipse in Standard Position, with horizontal  major axis
                                 
If the foci lie on the y-axis, then the standard equation is

= 1, where as before > .   
  

As before, a plot appears when you move the cursor to the end of the following block of blue code and press the Enter key.                                

            Plot of Ellipse in Standard Position, with vertical  major axis
    
3. Hyperbolas. A hyperbola is the set of all points P(x, y) in the plane the difference of whose distances from two fixed points (the foci) is a constant, 2a. If the foci lie on the x-axis, then the hyperbola has equation

= 1,
    

and asymptotes y = bx/a and y = --bx/a. The following code produces a picture.
          

            Plot of Hyperbola in Standard Position, with foci on x-axis
            
     If the foci lie on the y-axis, then the two branches have vertices on that axis, and the equation has the form

= 1.   

                                         
The following code generates a picture of the curve.                
    

                 Plot of Hyperbola in Standard Position, with foci on y-axis