2. Ellipse
The ellipse is the set of all points on
the plane for which the sum of
distances from two fixed points (foci)
is a positive constant.
3. Parts of an Ellipse:
Semi-axis - ½ the length of axis
Center - midpoint of the segment joining the foci
Vertices - endpoints of the major axis. The points on the ellipse which lie on the
line containing the foci
Major axis – The line segment joining the vertices, longer axis, contains foci
Co- Vertices – the points on the ellipse which are on the perpendicular bisector
of the major axis
Minor axis – The line segment joining the co-vertices, shorter axis
Latus Rectum - The line segment passing through the focus and perpendicular to
the major axis
Foci - two given fixed points on the major axis
Center Focus
Focus
5. Ellipse with horizontal major axis:
• Center is (0, 0).
• Length of major axis is 2a.
• Length of minor axis is 2b.
• Distance between center and either focus
is c with c2 = a2– b2, a > b > 0.
6. Ellipse with vertical major axis:
• Center is (0, 0).
• Length of major axis is 2a.
• Length of minor axis is 2b.
• Distance between center and either focus
is c with c2 = a2– b2, a > b > 0.
• LR = 2b2/a
• e = c/a
• d = a/e = a2/c
9. Ellipse with horizontal major axis:
• Center is (h, k).
• Length of major axis is 2a.
• Length of minor axis is 2b.
• Distance between center
• and either focus
is c with c2 = a2– b2, a > b
> 0.
10. Ellipse with vertical major axis:
• Center is (h, k).
• Length of major axis is 2a.
• Length of minor axis is 2b.
• Distance between center
and either focus
is c with c2 = a2– b2, a > b
> 0.
11. Pre-
Calculus
Ellipses with Center (h, k)
Standard
Equation
Focal Axis y = k x = h
Foci (h c, k) (h, k c)
Vertices (h a, k) (h, k a)
Semimajor Axis a a
Semiminor Axis b b
Pythagorean
Relation
a2 = b2 + c2 a2 = b2 + c2
2 2
2 2
(x h) (y k)
1
a b
2 2
2 2
(y k) (x h)
1
a b
12. ASSESSMENT
1. What is the standard form of the
ellipse that has vertices at (0, ±3) and foci
at ( 0, ±2)?
2. What is the standard form of equation
of the ellipse that has vertices (-2, -8) and
(-2, 2) and foci (-2, -7) and (-2, 1).?