Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)

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Identify the center of the ellipse as \( (h, k) = (-2, 3) \).

Since the major axis is vertical, the standard form of the ellipse equation is \( \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

Calculate the semi-major axis length \(a\) by dividing the major axis length by 2: \( a = \frac{10}{2} = 5 \).

Calculate the semi-minor axis length \(b\) by dividing the minor axis length by 2: \( b = \frac{4}{2} = 2 \).

Substitute \(h\), \(k\), \(a\), and \(b\) into the standard form equation: \( \frac{(x + 2)^2}{2^2} + \frac{(y - 3)^2}{5^2} = 1 \). This is the standard form of the ellipse equation.

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주요 개념

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Standard Form of an Ellipse Equation

The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis, the equation is ((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1, where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length.

Major and Minor Axes Lengths

The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. Their lengths are twice the semi-major (a) and semi-minor (b) axes, respectively. Knowing these lengths helps determine a and b values used in the ellipse equation.

Center of the Ellipse

The center (h, k) of the ellipse is the midpoint of both the major and minor axes. It shifts the ellipse from the origin to the point (h, k) in the coordinate plane, affecting the equation by translating x and y coordinates accordingly.