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Chapter 7: Problem 47

Graph each ellipse and give the location of its foci. $$\frac{(x-1)^{2}}{2}+\frac{(y+3)^{2}}{5}=1$$

Short Answer

Expert verified
The foci of the ellipse are located at \((1+\sqrt{3}, -3)\) and \((1-\sqrt{3}, -3)\).

Step by step solution

01

Identify the Center

The equation for this ellipse is in the format \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where (h, k) is the center of the ellipse. By comparing this with the given equation, we see that h=1 and k=-3. So, the center of the ellipse is at (1,-3).
02

Find the Lengths of the Axes

In the same equation, \(a^2\) and \(b^2\) are respectively the squares of the lengths of semi-major and semi-minor axes of the ellipse. The square root of the denominator under x is the semi-major axis length. The square root of the denominator under y is the semi-minor axis length. From the given equation, \(a^2 = 2\), and \(b^2 = 5\). Therefore, \(a = \sqrt{2}\) and \(b=\sqrt{5}\). The lengths of the major and minor axes are twice these values, so the major axis length is \(2\sqrt{5}\) and the minor axis length is \(2\sqrt{2}\).
03

Identify the Foci

The foci are located at a distance of \(c = \sqrt{b^2 - a^2}\) from the center along the major axis. Plugging in our known values, \(c = \sqrt{5 - 2} = \sqrt{3}\). Since the major axis of this ellipse is along the x-axis, the foci are at \((1+\sqrt{3}, -3)\) and \((1-\sqrt{3}, -3)\).
04

Sketch the Ellipse

To sketch the ellipse, first plot the center. Then, draw the major and minor axes, using the lengths calculated in Step 2. Next, plot the foci. Finally, sketch the ellipse shape around these components.

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Most popular questions from this chapter

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$8 y^{2}+4 x=0$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so I was able to graph the ellipse.

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

Describe how to graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)
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