Math Course Help With Ellipse

7.6 Ellipse:

7.6.1 Introduction: An ellipse is the locus of a point that moves in such a way that the ratio of its distance from a fixed point called the focus to its distance from a fixed line, called the directrix.

The eccentricity of a ellipse, denoted e, is defined as e = c/a, where c is half the distance between foci. Eccentricity is a number that describe the degree of roundness of the ellipse. For any ellipse, 0 ≤ e ≤ 1. The smaller the eccentricity, the rounder is the ellipse. If e = 0, it is a circle and F1, F2 are coincident. If e = 1, then it's a line segment, with foci at the two end points.

Vertexes of the ellipse are defined as the intersections of the ellipse and a line passing through foci. The distance between the vertexes is called major axis or focal axis. A line passing the center and perpendicular to the major axis is the minor axis. Half the length of major axis is called semi-major axis. Half the length of major axis is called semi-minor axis.

5.6.2 Standard equation of ellipse:

Recall that the equation of a circle centered at the origin has equation

x² + y² = r²

Where, r is the radius. Dividing by r2 we have

x2 + y2 =1
r2 r2

For an ellipse there are two radii, so that we can expect that the denominators should be different.

Hence we have the standard form of an ellipse centered at the origin:

x2 + y2 = 1
a2 b2

The points (a,0), (-a,0), (0,b), and (0,-b) are called the vertices of the ellipse.

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