Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. 9 x^2+4

step 1 Hello friends we have to find the center vertices and the focus and the eccentricity of the give

Find the center, foci, vertices, and eccentricity of the...

Key Concepts

Ellipse Standard Form

An ellipse in standard form is expressed as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) represents the center, and a and b are the lengths of the semi-major and semi-minor axes, respectively. This form simplifies the process of identifying all key features of the ellipse, such as the vertices, foci, and orientation.

Completing the Square in Conic Sections

Completing the square is an algebraic technique used to convert a general quadratic equation into a standard form that is easier to analyze. In the context of ellipses, this method helps isolate the squared terms for both x and y, thereby revealing the center and the lengths of the axes required to describe the ellipse's geometry.

Center of the Ellipse

The center of an ellipse is the point (h, k) around which the ellipse is symmetrically arranged. It is found directly from the completed square form of the equation and serves as the origin for measuring distances to the vertices and foci.

Vertices of the Ellipse

Vertices are the points on the ellipse that lie along the major axis, representing the maximum distances from the center in that direction. They are determined by adding and subtracting the semi-major axis length from the appropriate coordinate of the center.

Foci of the Ellipse

The foci are two fixed points along the major axis of the ellipse, located at a distance c from the center, where c is found using the relationship c² = a² - b². These points are significant because any point on the ellipse maintains a constant sum of distances to them, a defining property of the ellipse.

Eccentricity

Eccentricity, defined as e = c/a for an ellipse, measures how much the ellipse deviates from a perfect circle (which has an eccentricity of 0). It is a critical parameter that describes the shape's elongation, with values between 0 and 1 indicating an ellipse, where values closer to 1 denote a more stretched shape.

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