Problem 33 Find an equation of the followin... [FREE SOLUTION]

Chapter 11: Problem 33

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse whose major axis is on the \(x\) -axis with length 8 and whose minor axis has length 6

Short Answer

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Answer: The equation of the ellipse is \(\frac{x^2}{16}+\frac{y^2}{9}=1\). The vertices are located at points \((-4,0), (4,0), (0,-3), (0,3)\), and the foci are at points \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\).

Step by step solution

Identify the lengths of the semi-major and semi-minor axes

First, we need to find the lengths of the semi-major (a) and semi-minor axes (b) by dividing the given lengths of the major and minor axes by 2, respectively. \(a = \frac{8}{2}=4\) \(b = \frac{6}{2}=3\)

Write the standard equation of the ellipse

The standard equation of an ellipse centered at the origin (0, 0) is given by: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) Using the values of the semi-major and semi-minor axes obtained in step 1, the equation of the given ellipse is: \(\frac{x^2}{4^2}+\frac{y^2}{3^2}=1\) Final equation: \(\frac{x^2}{16}+\frac{y^2}{9}=1\)

Identify the coordinates of the vertices

The vertices are the points where the ellipse intersects the major and minor axes. Since the major axis is along the \(x\)-axis, the vertices are at the points \((\pm a, 0)\). The minor axis is along the \(y\)-axis, so the vertices are also at the points \((0, \pm b)\). Vertices: \((-4,0), (4,0), (0,-3), (0,3)\)

Calculate the distance between the center and the foci

Now, we need to find the focal distance (c) based on the lengths of the semi-major and semi-minor axes: \(c = \sqrt{a^2 - b^2} = \sqrt{4^2 - 3^2} = \sqrt{7}\)

Identify the coordinates of the foci

Since the major axis is along the \(x\)-axis, the foci are located at the points \((\pm c, 0)\) from the center (0, 0). Foci: \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\)

Sketch a graph of the ellipse

Plot the ellipse with its equation \(\frac{x^2}{16}+\frac{y^2}{9}=1\). Label the vertices and foci as we calculated their coordinates. Vertices: \((-4,0), (4,0), (0,-3), (0,3)\) Foci: \((-\sqrt{7}, 0)\) and \((\sqrt{7}, 0)\) Draw the graph using the information above. The center of the ellipse will be at the origin (0, 0). Plot the vertices along the major and minor axes, and the foci along the major axis.

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91影视

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse whose major axis is on the \(x\) -axis with length 8 and whose minor axis has length 6

Short Answer

Step by step solution

Identify the lengths of the semi-major and semi-minor axes

Write the standard equation of the ellipse

Identify the coordinates of the vertices

Calculate the distance between the center and the foci

Identify the coordinates of the foci

Sketch a graph of the ellipse

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