Graph each ellipse and locate the foci. x2/9 +y2/36= 1

Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 3

Chapter 8, Problem 3

Graph each ellipse and locate the foci. x²/9 +y²/36= 1

Verified step by step guidance

Identify the standard form of the ellipse equation given: \(\frac{x^{2}}{9} + \frac{y^{2}}{36} = 1\). Here, \(a^{2}\) and \(b^{2}\) are the denominators under \(x^{2}\) and \(y^{2}\) respectively.

Determine which denominator is larger to identify the major axis. Since \(36 > 9\), the major axis is vertical, and \(a^{2} = 36\), so \(a = 6\). The minor axis corresponds to \(b^{2} = 9\), so \(b = 3\).

Plot the ellipse centered at the origin \((0,0)\) with vertices along the major axis at \((0, \pm a)\), which are \((0, \pm 6)\), and co-vertices along the minor axis at \((\pm b, 0)\), which are \((\pm 3, 0)\).

Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values to find \(c^{2} = 36 - 9\).

Locate the foci on the major axis at points \((0, \pm c)\), which are \((0, \pm \sqrt{c^{2}})\). These points lie inside the ellipse along the vertical axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 (horizontal or vertical).

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 understand the shape and size of the ellipse based on its equation.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse along the major axis, found using c² = |a² - b²|, where c is the distance from the center to each focus. Knowing the foci is essential for understanding ellipse properties, such as the sum of distances from any point on the ellipse to the foci being constant.

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Foci and Vertices of an Ellipse

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