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Ahhhhh

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Submitted By ellimakgal
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Definition of Ellipse

Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a.

General Equation of the Ellipse

From the general equation of all conic sections, [pic] and [pic] are not equal but of the same sign. Thus, the general equation of the ellipse is

[pic]

or [pic]

Standard Equations of Ellipse

Elements of the ellipse are shown in the figure above.

1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Semi-major axis = a and semi-minor axis = b. 3. Location of foci c, with respect to the center of ellipse. [pic]. 4. Length latus rectum, LR 5. Consider the right triangle F1QF2:

Based on the definition of ellipse: [pic]

[pic]

[pic]

By Pythagorean Theorem [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity.

6. Eccentricity, e

DEFINITION: Eccentricity of Conic

Eccentricity is a measure of how much a conic deviate from being circular, making the eccentricity of the circle obviously equal to zero. It is the ratio of focal distance to directrix distance of the conic section.

[pic]

From the figure of the ellipse above, [pic]

From [pic]

[pic]

[pic]

Thus, [pic]

[pic]

7. Location of directrix d, with respect to the center of ellipse.

From the derivation of eccentricity,

[pic]
TABLE OF PROPERTIES
|STANDARD FORM OF EQUATION |VERTICES |FOCI |ENDPOINTS OF THE |ENDPOINTS OF THE MINOR |EQUATION OF THE MAJOR |
| | | |LATERA RECTA |AXIS |AXIS |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |

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