1st Lesson Conic Sections

1st Lesson: Conic Sections

The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9

GENERAL EQUATION
Ax²­+Bxy+Cy²+Dx+Ey+F=0


THE DISCRIMINANT FORMULA FOR CONIC SECTION

 b² – 4ac > 0 HYPERBOLA
b²  - 4ac = 0 PARABOLA
b²  - 4ac <  0 CIRCLE

         where A = C B = 0
b²  - 4ac <  0 ELLIPSE
         where A≠c B=0

EXAMPLES:
1. G.E
 2y² + 20x + 2y – 1 = 0
 A= 0  B=  0  C= 2
b² – 4ac = (0)⁰ – 4 (0)(2)
                = 0 – 0
                = 0 PARABOLA

S.F : (h,k) (y-k)² = 4a(x-h)

2y² + 20x + 12y – 1 = 0

2y² + 12y       20x – 1 = 0

2(y² + 6y + 9) = -20x + 1

 2(y+3)²  = -20x + 1 + 9

    2

(y+3)² = -10x + 10

S.F = (y+3)² = -10x + 10

h = 5 k = 3

 2. G.E

6x² – 5y + 36x + 20y – 16 = 0

A= 6  B= 0  C= -5

b² – 4ac = (0)² – 4 (6)(-5)

= 120 < 0  HYPERBOLA

S.F

6x² + 36x ____ -5y + 20y – 16 = 0

6(x²+6x+9) – 5(y² – 4y + 4)= 16 + 9 + 4

6(x+3)² – 5(y-2)² = 29

              30

(x +3)² – (y-2)²

     5            6

   10          10

(h = 3) (k= -2)