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.
$$
\text { Eccentricity: } \sqrt{5} / 3, \text { foci on } x \text { -axis, length of major axis: } 12
$$

Key Concepts

Ellipse

An ellipse is a geometric shape and a type of conic section defined as the set of all points in a plane such that the sum of the distances from two fixed points (called the foci) is constant. It exhibits symmetry and has distinct major and minor axes.

Standard Equation of an Ellipse

The standard form of an ellipse’s equation, typically centered at the origin, is given as x²/a² + y²/b² = 1, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively. This form is used to easily identify the ellipse's geometric properties.

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being circular. It is defined as the ratio of the distance from the center to a focus (c) over the length of the semi-major axis (a), with values between 0 (a circle) and 1 (a highly elongated ellipse).

Foci and Their Relationship to the Ellipse

The foci are two fixed points located along the major axis of an ellipse. Their positions determine the ellipse’s shape and are related to the semi-major and semi-minor axes through the equation c² = a² - b², linking them to the eccentricity.

Major Axis

The major axis is the longest diameter of an ellipse, passing through the center and both foci. Its length, which is twice the value of the semi-major axis, is a fundamental parameter used to define the size and shape of the ellipse.

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