Ch. 7 - Conic Sections

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

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 45

Chapter 8, Problem 45

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

Verified step by step guidance

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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Key Concepts

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

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.

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Related Practice

Textbook Question

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x² - 2x - 4y + 9 =0

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Textbook Question

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. y² - 2y + 12x - 35 = 0

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Textbook Question

In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $16 x^{2} - y^{2} + 64 x - 2 y + 67 = 0$

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Textbook Question

Convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $x^{2} - y^{2} - 2 x - 4 y - 4 = 0$

Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1

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