Finding the Equation of an Ellipse Find an equation for the...

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions.
$$
\text { Eccentricity: } \sqrt{3} / 2, \text { foci on } y \text { -axis, length of major axis: } 4
$$

Key Concepts

Relationship Between a, b, and c

In an ellipse, the relationship between the semi-major axis 'a', semi-minor axis 'b', and the distance 'c' from the center to a focus is given by the equation c² = a² - b². This formula connects the geometric dimensions of the ellipse with its foci and is fundamental in solving for unknown parameters given certain conditions.

Eccentricity

Eccentricity is a parameter that quantifies the deviation of the ellipse from a perfect circle, defined as the ratio c/a, where 'c' is the distance from the center to each focus and 'a' is the semi-major axis. For any ellipse, this value lies between 0 and 1, with values closer to 0 indicating a more circular shape and those closer to 1 indicating a more elongated shape.

Foci

The foci are two fixed points located symmetrically with respect to the center of the ellipse along its major axis. Their positions are critical in defining the ellipse, as every point on the ellipse maintains a constant sum of distances to these points.

Standard Equation of an Ellipse

The standard equation of an ellipse is expressed in a form such as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse, 'a' is the semi-major axis, and 'b' is the semi-minor axis. The orientation of the ellipse (whether its major axis is horizontal or vertical) is determined by the relative positions of a and b.

Ellipse

An ellipse is a conic section defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. This geometric property results in an oval shape that can be oriented horizontally or vertically depending on the placement of its foci.

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