Summary 3: the Ellipse

Definition-An ellipse is the set of points in a plane such that the sum of the distances to two fixed points called foci, is constant. The constant sum is the major diameter of the ellipse.  Theorem 1: The equation of the ellipse with center at (0, 0) , foci at (c, 0) and (-c,0) and major diameter 2a is given by x2 y2  1 where b2  a2  c2 a2 b2 2b is called the minor diameter of the ellipse.

 Theorem 2: The equation of the ellipse with center at (0, 0) , foci at (0, c) and (0,-c) and major diameter 2a is given by x2 y2  1 where b2  a2  c2 b2 a2 2b is the minor diameter x2 y2  Theorem 3. The ellipse  1 where a  b, b2  a2  c2 is the conic a2 b2 a2 c with foci at  c, 0. directrix x   and eccentricity e  1. The b a  2   b  ends of the latera recta are given by  c,  .  a  x2 y2 Theorem 4. The ellipse  1 where a  b, b2  a2  c2 is the conic b2 a2 a2 c with foci at (0,  c. directrix y   and eccentricity e  1. The ends b a  2   b  of the latera recta are given by   , c.  a 

Note: a circle is a degenerate form of an ellipse where a = b = r.

Theorem 5: the eccentricity of a circle is 0. Proof a circle is an ellipse  c2  a2  b2  0 because a  b c 0  e    0 a a

1 Summary of formulas:

STANDARD FORMS OF AN ELLIPSE: c 2  a 2  b 2 , a ≥ b and c < a

Ends of Equation Center Vertices Co-vertices foci latera recta Major diameter on the x-axis b 2 (  c ,  ) x 2 y 2 a   1 (0,0) (±a,0) (0,±b) (±c,0) 2 b 2 a 2 b 2 length= a Major diameter =2a

Minor diameter=2b b 2 Major diameter on the y-axis (0,0) (  ,  c ) x 2 y 2 (0,±a) ±b,0) (0,±c) a   1 2 b 2 b 2 a 2 length a Major diameter =2a

Minor diameter=2b

Y Y Directrix

a c Directrix directrix b

-b b x-axis -a -c c a x

-b -c -a directrix

Major Diameter on y-axis Major Diameter on x-axis

Translation formulas: x  h2 y  k2 x2 y2  1 is the equation of the ellipse  1 a2 b2 a2 b2 translated h units horizontally and k units vertically. x  h2 y  k2 x2 y2  1 is the equation of the ellipse  1 b2 a2 b2 a2 translated h units horizontally and k units vertically.

The axes x  x  h and y  y  k are called the translation axis.