Ch. 7 - Conic Sections

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

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 47

Chapter 8, Problem 47

Graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 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 $\frac{(x - 2)^2}{7} + \frac{(y + 1)^2}{11} = 1$, determine the center of the ellipse as $(2, -1)$.

Compare the denominators to find $a^2$ and $b^2$. Since $11 > 7$, set $a^2 = 11$ and $b^2 = 7$. This means the major axis is vertical.

Calculate the distance $c$ from the center to each focus using the formula $c = \sqrt{a^2 - b^2}$. Substitute the values to find $c = \sqrt{11 - 7}$.

Locate the foci along the major axis (vertical axis) by adding and subtracting $c$ from the $y$-coordinate of the center. The foci are at $(2, -1 + c)$ and $(2, -1 - c)$.

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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

An ellipse can be expressed in the standard form $ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 $, where $(h, k)$ is the center. The denominators $a^2$ and $b^2$ represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Identifying these values helps in graphing the ellipse accurately.

Determining the Orientation of the Ellipse

The larger denominator between $a^2$ and $b^2$ indicates the major axis direction: if $a^2 > b^2$, the ellipse is stretched horizontally; if $b^2 > a^2$, it is stretched vertically. This orientation is crucial for locating the foci and sketching the ellipse.

Finding the Foci of an Ellipse

The foci lie along the major axis, located at a distance $c$ from the center, where $c = \sqrt{|a^2 - b^2|}$. Knowing $c$ and the center coordinates allows you to find the exact positions of the foci, which are key points defining the ellipse's shape.

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Foci and Vertices of an Ellipse

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