Ellipse

In: Computers and Technology

Submitted By ilovemeame
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The Ellipse
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…...

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