Graphing Shifted Ellipses An equation of an elllipse is...

Graphing Shifted Ellipses An equation of an elllipse is given. (a) Find the center, and foci of the ellipse.
(b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse.
$$
9 x^{2}-54 x+y^{2}+2 y+46=0
$$

Key Concepts

Foci of the Ellipse

The foci are two fixed points along the major axis of the ellipse and are crucial to its geometric definition. Their positions are determined by the relationship between the distances 'a' and 'b' (with c^2 = a^2 - b^2 for a horizontal ellipse, for example). Locating the foci is important both for understanding the intrinsic properties of the ellipse and for various applications in physics and engineering.

Graphing Ellipses

Graphing an ellipse involves plotting its center, vertices, co-vertices, and foci, and sketching a smooth curve through these points. This process not only requires the identification of key parameters like the center, axes lengths, and foci but also involves understanding the overall structure of the ellipse to produce an accurate representation on the coordinate plane.

Major and Minor Axes

The lengths and orientations of the major and minor axes reveal the shape and size of the ellipse. The major axis is the longest diameter through the center, while the minor axis is the shortest. Their lengths are typically represented using the parameters 'a' and 'b' in the standard form, and knowing these helps in determining the ellipse's overall geometry.

Completing the Square

Completing the square is a vital algebraic technique used to convert quadratic expressions into a perfect square form. This step is essential in rewriting the ellipse's equation into its standard form, as it helps in isolating squared terms and identifying the translated center of the ellipse.

Standard Form of an Ellipse

This concept involves rewriting the equation of an ellipse into its standard form, which clearly shows the geometric properties such as the center, the lengths of the major and minor axes, and the orientation of the ellipse. The process often involves algebraic manipulations (like factoring and completing the square) to transform the equation into a format where these parameters are readily identifiable.

Center of the Ellipse

The center of an ellipse is the midpoint of its major and minor axes. In the standard form of an ellipse, the center appears as the coordinates (h, k), which represent how much the ellipse has been shifted from the origin. Determining the center is fundamental for graphing and understanding other properties of the ellipse.

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