An ellipse with center (-4,-2) has a horizontal major axis...

An ellipse with center $(-4,-2)$ has a horizontal major axis of length 6 and a minor axis of length 4 . Which equation represents the ellipse? A. $\frac{(x-4)^{2}}{4}-\frac{(y-2)^{2}}{9}=1$ B. $\frac{(x+2)^{2}}{9}-\frac{(y+4)^{2}}{4}=1$ C. $\frac{(x+4)^{2}}{9}+\frac{(y+2)^{2}}{4}=1$ D. $\frac{(x+4)^{2}}{4}+\frac{(y+2)^{2}}{9}=1$

An ellipse with center $(-4,-2)$ has a horizontal major axis of length 6 and a minor axis of length 4 . Which equation represents the ellipse?
A. $\frac{(x-4)^{2}}{4}-\frac{(y-2)^{2}}{9}=1$
B. $\frac{(x+2)^{2}}{9}-\frac{(y+4)^{2}}{4}=1$
C. $\frac{(x+4)^{2}}{9}+\frac{(y+2)^{2}}{4}=1$
D. $\frac{(x+4)^{2}}{4}+\frac{(y+2)^{2}}{9}=1$

Key Concepts

Orientation of the Ellipse

The orientation of an ellipse indicates whether its major axis is horizontal or vertical. When the major axis is horizontal, the larger value (a²) appears under the (x - h)² term in the standard equation; if the major axis is vertical, it appears under the (y - k)² term. This distinction determines the geometric layout of the ellipse in the coordinate plane.

Semi-major and Semi-minor Axes

The semi-major axis (a) is half the length of the major axis, and the semi-minor axis (b) is half the length of the minor axis. These values are critical for determining the denominators in the ellipse's equation and are used to calculate the horizontal and vertical stretches of the ellipse.

Center of an Ellipse

The center of an ellipse is the point (h, k) from which the ellipse is symmetrically expanded in all directions. In the standard form, the terms (x - h) and (y - k) indicate that the ellipse has been translated from the origin to this center, and this point is used as a reference for further calculations of distances along the axes.

Major and Minor Axes

The major axis of an ellipse is its longest diameter, extending through the center and defining the widest part of the ellipse, while the minor axis is the shortest diameter, perpendicular to the major axis at the center. The lengths of these axes are double the lengths of the semi-major and semi-minor axes respectively, and these values directly impact the shape and spread of the ellipse.

Standard Equation of an Ellipse

The standard equation of an ellipse that is oriented along the coordinate axes is given by ((x - h)²)/a² + ((y - k)²)/b² = 1 when the major axis is horizontal, or by ((x - h)²)/b² + ((y - k)²)/a² = 1 when the major axis is vertical. Here, (h, k) represents the center of the ellipse, and a and b are the semi-major and semi-minor axes respectively.

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