(a) find the standard form of the equation of the ellipse,...

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph.
$$9 x^{2}+25 y^{2}-36 x-50 y+61=0$$

Key Concepts

Eccentricity

Eccentricity is a measure of how 'stretched' an ellipse is, defined as the ratio c/a where c is the distance from the center to one of the foci, and a is the length of the semimajor axis. It is a number between 0 and 1, with values closer to 0 corresponding to a more circular ellipse and values closer to 1 indicating a more elongated shape.

Foci

Foci are two fixed points located along the major axis of the ellipse. The sum of the distances from any point on the ellipse to the foci is constant. The distance from the center to a focus is found using the relationship c² = a² - b² for an ellipse, and they play a key role in the geometric definition of an ellipse.

Vertices

Vertices are the points on the ellipse that lie along the major axis at the maximum distance from the center. They are determined by the values of a or b in the standard form equation, depending on the orientation of the ellipse, and mark the endpoints of the ellipse along that axis.

Completing the Square

A method used to transform a quadratic equation into a form that makes the relationships between variables clearer, especially when rewriting conic section equations into their standard form. This process groups and factors quadratic and linear terms so that the expression can be written as a squared term plus or minus a constant, facilitating further analysis.

Standard Form of an Ellipse

The standard form for the equation of an ellipse is written as ((x-h)²/a²) + ((y-k)²/b²) = 1, where (h, k) is the center of the ellipse and a and b represent the lengths of the semimajor and semiminor axes, respectively. This form makes it straightforward to identify the key attributes of the ellipse.

Graphing Ellipses

Graphing an ellipse involves plotting its center, vertices, foci, and drawing a smooth, symmetric curve that represents all points satisfying the ellipse equation. Using graphing utilities can help verify the calculated parameters and provide a visual understanding of the ellipse's geometry.

Center of the Ellipse

The center is the midpoint of the ellipse and is found after rewriting the equation in standard form. It is crucial in determining the symmetry of the ellipse and serves as the reference point from which other features like vertices and foci are measured.

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