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Words 520

Pages 3

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, A and C are not equal but of the same sign. Thus,the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or

Standard Equations of Ellipse

From the figure above, and

From the definition above,

Square both sides

Square again both sides

From triangle OV3F2 (see figure above)

Thus,

Divide both sides by a2b2

The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above.

Ellipse with center at the origin

Ellipse with center at the origin and major axis on the x-axis.

Ellipse with center at the origin and major axis on the y-axis.

Ellipse with center at (h, k)

Ellipse with center at (h, k) and major axis parallel to the x-axis.

Ellipse with center at (h, k) and major axis parallel to the y-axis.

The Hyperbola

Submitted by Romel Verterra on February 21, 2011 - 1:33pm

Definition

Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a. General Equation

From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.) or

Standard Equations

From the definition:

From the figure:

Thus,

The equation we just derived above is the standard equation of hyperbola with center at the origin and transverse axis on the x-axis (see figure above). Below are the four standard equations of hyperbola. The first equation is the one we derived just derived. Hyperbola with center at the origin

Hyperbola with center at the origin and transverse axis on the x-axis.

Hyperbola with center at the origin and transverse axis on the y-axis.

Hyperbola with center at any point (h, k)

Hyperbola with center at (h, k) and transverse axis parallel to the x-axis.

Hyperbola with center at (h, k) and transverse axis parallel to the y-axis.

Elements of Hyperbola 1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Transverse axis = 2a and conjugate axis = 2b 3. Location of foci c, relative to the center of hyperbola.

4. Latus rectum, LR

5. Eccentricity, e

The eccentricity of hyperbola is always greater than one.

6. Location of directrix d relative to the center of hyperbola. or 7. Equation of asymptotes.

where m is (+) for upward asymptote and m is (-) for downward. m = b/a if the transverse axis is horizontal and m = a/b if the transverse axis is vertical…...

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