Table of standard ellipse
| Equation | x² /a² +y² / b² =1 when a > b
c²= a² – b² TYPE 1 ↓ |
x² /b² +y² / a² =1 when a > b
c²= a² – b² TYPE 2 ↓ |
| Focus | (± C, 0) | (± C, 0) |
| Major axis | y= 0 | x=0 |
| Directrices | x = ± c / e² | y = ± c / e² |
| Vertices | (± a, 0) | (0, ± a) |
| Co vertices | (0, ± b) | (± b, 0) |
| Center | (0, 0) | (0, 0) |
| Eccentricity | e= c / a < 1 | e= c / a < 1 |
Graph of an Ellipse Type 1 and 2
Let the equation of the ellipse be
x² /a² +y² / b² =1
since only even powers of both x and y occur in the figure, The cure is symmetric with respect to
both the axis
From the figure, we note that
x² /a² ≤ 1 and y² / b² ≤ 1
that is
x² ≤ a² and y² ≤ b²
-a ≤ x ≤ a and -b ≤ x ≤ b
thus, all the points or within the rectangle. The curve meets the x-axis at A (-a, 0) and
it meets the y-axis at B(0, -b) B'(0, b)
The graph of the ellipse is
x² /b² +y² / a² =1 a > b
TABLE IMPORTANT NOTE STANDARD ELLIPSE
| (1) each ellipse have length of major axis = 2a
(2) each ellipse of minor axis = 2b (3) each ellipse length of Latusrectum = 2b² / a (4) Foci lies on the major axis |
Example:
Find an equation of ellipse having center at (0, 0) focus at (0, -3) and one vertex at (0,4)Example :
Solution:
length of the major axis is
a = 4
foci = F= 3
from b² = a² – c ²
b² = 16 – 9
b² = 7
b = √ 7 which is the length of semi minor axis
since the Foci lies on the y-axis equation
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