Graph each ellipse and locate the foci. x2/25 +y2/64 = 1

Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 5

8장, 문제 5

Graph each ellipse and locate the foci. x²/25 +y²/64 = 1

검증된 단계별 안내

Identify the standard form of the ellipse equation given: \(\frac{x^{2}}{25} + \frac{y^{2}}{64} = 1\). Here, \(a^{2}\) and \(b^{2}\) are the denominators under \(x^{2}\) and \(y^{2}\) respectively.

Determine which denominator is larger to find the major axis. Since \(64 > 25\), the major axis is vertical, so \(a^{2} = 64\) and \(b^{2} = 25\).

Calculate the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{64}\) and \(b = \sqrt{25}\).

Find the distance \(c\) from the center to each focus using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values of \(a^{2}\) and \(b^{2}\) to find \(c\).

Plot the ellipse centered at the origin with vertices at \((0, \pm a)\) and co-vertices at \((\pm b, 0)\). Then, mark the foci at \((0, \pm c)\) along the major axis.

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주요 개념

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Standard Form of an Ellipse

The equation x²/a² + y²/b² = 1 represents an ellipse centered at the origin. Here, a² and b² are the denominators under x² and y², indicating the lengths of the semi-major and semi-minor axes. Identifying which denominator is larger helps determine the ellipse's orientation.

Graphing an Ellipse

To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connect these points smoothly to form the ellipse. This visual representation helps in understanding the shape and size of the ellipse.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse along the major axis. Their distance from the center is c, found using c² = |a² - b²|. Knowing c allows you to place the foci accurately, which is essential for understanding ellipse properties like reflection and eccentricity.