Graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1

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Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.

From the given equation \(\frac{(x + 1)^2}{36} + \frac{(y - 4)^2}{4} = 1\), rewrite the terms to identify the center: \(h = -1\) and \(k = 4\).

Determine the values of \(a^2\) and \(b^2\): here, \(a^2 = 36\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is horizontal.

Calculate the lengths of the semi-major axis \(a = \sqrt{36} = 6\) and the semi-minor axis \(b = \sqrt{4} = 2\).

Find the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Then, locate the foci at \((h \pm c, k)\) along the horizontal axis.

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주요 개념

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Standard Form of an Ellipse

The standard form of an ellipse equation is given by (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center. The denominators a² and b² represent the squares of the ellipse's semi-major and semi-minor axes, respectively. Understanding this form helps in identifying the ellipse's size, shape, and position on the coordinate plane.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant. Their locations depend on the values of a and b, with the distance from the center to each focus given by c = √(a² - b²) when a > b. Knowing how to find the foci is essential for graphing and understanding ellipse properties.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, co-vertices, and foci. The vertices lie along the major axis at a distance a from the center, while the co-vertices lie along the minor axis at a distance b. Accurate graphing requires understanding the orientation of the ellipse based on which denominator is larger.