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Chapter 7: Problem 40

Graph each ellipse and give the location of its foci. $$(x-3)^{2}+9(y+2)^{2}=18$$

Short Answer

Expert verified
The locations of the foci are (-1, -2) and (7, -2). The graph will be an ellipse centered at (3, -2) with a vertical minor axis of length \(\sqrt{2}\) and a horizontal major axis of length \(\sqrt{18}\).

Step by step solution

01

Standardize the equation

Firstly standardize the given equation to the form of an ellipse equation. In our case divide each side of the given equation by 18 to get \((x-3)^2/18 + (y+2)^2/2 = 1\)
02

Identify the center, semi-major and semi-minor axes

Comparing with standard ellipse equation, it's clear the center of the ellipse is at point (3, -2). Also, \(a^2 = 18\) and \(b^2 = 2\), so, the semi-major axis (a) is \(\sqrt{18}\) and the semi-minor axis (b) is \(\sqrt{2}\).
03

Graph the ellipse

Plot the center at (3, -2) on a graph. Draw the major axes as a horizontal line through the center having a length of 2a = 2 * \(\sqrt{18}\), and minor axes as a vertical line through the center having a length of 2b = 2 * \(\sqrt{2}\). Then draw the ellipse around these axes.
04

Find the foci

The foci of an ellipse are located on the major axis at a distance of \(\sqrt{a^2 - b^2}\) from the center. Substituting the values for a and b gives \(\sqrt{18 - 2} = 4\). So, the foci are located at (3-4, -2) and (3+4, -2), or (-1, -2) and (7, -2). You can denote these points on your graph to complete it.

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Most popular questions from this chapter

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