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Chapter 7: Problem 37

Graph each ellipse and give the location of its foci. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

Short Answer

Expert verified
The center of the ellipse is at (2, 1), and it has a semi-major axis of 3 along the x-axis and a semi-minor axis of 2 along the y-axis. The foci of the ellipse are located at \((2 - \sqrt{5}, 1)\) and \((2 + \sqrt{5}, 1)\).

Step by step solution

01

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

The standard form of an ellipse equation gives us the center, semi-major axis and semi-minor axis directly. In this case, the center is at (h, k) = (2, 1). The semi-major axis \(a\) is the square root of the larger denominator and semi-minor axis \(b\) is the square root of the smaller denominator, so \(a = \sqrt{9} = 3\) and \(b = \sqrt{4} = 2\). The semi-major axis is along the x-axis and the semi-minor axis is along the y-axis.
02

Find the location of the foci

The foci are located along the major axis, a distance of \(\sqrt{a^2 - b^2}\) from the center. This gives us \(\sqrt{9 - 4} = \sqrt{5}\). Therefore, the foci are located at \((2 - \sqrt{5}, 1)\) and \((2 + \sqrt{5}, 1)\).
03

Sketch the ellipse

Mark the center of the ellipse at (2, 1) on a graph. From there, draw the semi-major axis along the x-axis 3 units in both directions, and the semi-minor axis along the y-axis 2 units in both directions. Sketch the ellipse that touches the ends of these axes. Mark the foci inside the ellipse.

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Most popular questions from this chapter

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(y-2)^{2}-(x+3)^{2}=5$$

Graph each ellipse and give the location of its foci. $$\frac{(x-3)^{2}}{9}+\frac{(y+1)^{2}}{16}=1$$

Which one of the following is true? a. If one branch of a hyperbola is removed from a graph, then the branch that remains must define \(y\) as a function of \(x .\) b. All points on the asymptotes of a hyperbola also satisfy the hyperbola's equation. c. The graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) does not intersect the line \(y=-\frac{2}{3} x\) d. Two different hyperbolas can never share the same asymptotes.

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\).

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$25 x^{2}+4 y^{2}=100$$
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