About Conic Sections: The Ellipse:
An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. The two fixed points are known as foci. The problems in this section involve finding the standard form of the ellipse and graphing the ellipse. We will also be asked to find the vertices and foci of the ellipse.
Test Objectives
- Demonstrate the ability to find the foci for an ellipse
- Demonstrate the ability to find the vertices for an ellipse
- Demonstrate the ability to write the equation of an ellipse
- Demonstrate the ability to graph an ellipse
#1:
Instructions: Sketch the graph, and state the center, vertices, covertices, and foci.
$$a)\hspace{.2em}\frac{x^2}{36}+ \frac{(y + 2)^2}{4}=1$$
$$b)\hspace{.2em}\frac{(x + 1)^2}{16}+ \frac{(y - 1)^2}{25}=1$$
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#2:
Instructions: Sketch the graph, and state the center, vertices, covertices, and foci.
$$a)\hspace{.2em}16x^2 + 49y^2 + 98y - 735=0$$
$$b)\hspace{.2em}36x^2 + 16y^2 - 36x - 16y - 563=0$$
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#3:
Instructions: Sketch the graph and write the equation in standard form.
$$a)\hspace{.2em}\text{vertices:} \, (-3, -10), (-13, -10)$$ $$\text{foci:} \, (-4, -10), (-12, -10)$$
$$b)\hspace{.2em}\text{vertices:} \, (-2, 23), (-2, -3)$$ $$\text{foci:} \, (-2, 15), (-2, 5)$$
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#4:
Instructions: Sketch the graph and write the equation in standard form.
$$a)\hspace{.2em}\text{vertices:} \, (7, 8), (7, -2)$$ $$\text{foci:} \, (7, 7), (7, -1)$$
$$b)\hspace{.2em}\text{vertices:} \, (13, 9), (3, 9)$$ $$\text{foci:} \, (11, 9), (5, 9)$$
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#5:
Instructions: Sketch the graph and write the equation in standard form.
$$a)\hspace{.2em}\text{vertices:} \, (0, 3), (0, -7)$$ $$\text{foci:} \, (0, 1), (0, -5)$$
$$b)\hspace{.2em}\text{vertices:} \, (1, 8), (1, -2)$$ $$\text{foci:} \, (1, 7), (1, -1)$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$\text{center:} \, (0, -2)$$ $$\text{vertices:} \, (6, -2), (-6, -2)$$ $$\text{covertices:} \, (0, -4), (0, 0)$$ $$\text{foci:} \, (4 \sqrt{2}, -2), (-4\sqrt{2}, -2)$$
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$$b)\hspace{.2em}$$ $$\text{center:} \, (-1, 1)$$ $$\text{vertices:} \, (-1, 6), (-1, -4)$$ $$\text{covertices:} \, (-5, 1), (3, 1)$$ $$\text{foci:} \, (-1, 4), (-1, -2)$$
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$\text{center:} \, (0, -1)$$ $$\text{vertices:} \, (7, -1), (-7, -1)$$ $$\text{covertices:} \, (0, -5), (0, 3)$$ $$\text{foci:} \, (\sqrt{33}, -1), (-\sqrt{33}, -1)$$
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$$b)\hspace{.2em}$$ $$\text{center:} \, \left(\frac{1}{2}, \frac{1}{2}\right)$$ $$\text{vertices:} \, \left(\frac{1}{2}, \frac{13}{2}\right), \left(\frac{1}{2}, -\frac{11}{2}\right)$$ $$\text{covertices:} \, \left(-\frac{7}{2}, \frac{1}{2}\right), \left(\frac{9}{2}, \frac{1}{2}\right)$$ $$\text{foci:} \, \left(\frac{1}{2}, \frac{4\sqrt{5}+ 1}{2}\right), \left(\frac{1}{2}, \frac{-4 \sqrt{5}+ 1}{2}\right)$$ Note (for the graph below): $$\frac{4\sqrt{5}+ 1}{2} ≈ 4.972$$ $$\frac{-4 \sqrt{5}+ 1}{2} ≈ -3.972$$
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#3:
Solutions:
$$a)\hspace{.2em}\frac{(x + 8)^2}{25}+ \frac{(y + 10)^2}{9}=1$$ $$\text{center:} \, \left(-8, -10\right)$$ $$\text{vertices:} \, \left(-3, -10\right), \left(-13, -10\right)$$ $$\text{covertices:} \, \left(-8, -13\right), \left(-8, -7\right)$$ $$\text{foci:} \, \left(-4, -10\right), \left(-12, -10\right)$$
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$$b)\hspace{.2em}\frac{(x + 2)^2}{144}+ \frac{(y - 10)^2}{169}=1$$ $$\text{center:} \, \left(-2, 10\right)$$ $$\text{vertices:} \, \left(-2, 23\right), \left(-2, -3\right)$$ $$\text{covertices:} \, \left(-14, 10\right), \left(10, 10\right)$$ $$\text{foci:} \, \left(-2, 15\right), \left(-2, 5\right)$$
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#4:
Solutions:
$$a)\hspace{.2em}\frac{(x - 7)^2}{9}+ \frac{(y - 3)^2}{25}=1$$ $$\text{center:} \, \left(7, 3\right)$$ $$\text{vertices:} \, \left(7, 8\right), \left(7, -2\right)$$ $$\text{covertices:} \, \left(4, 3\right), \left(10, 3\right)$$ $$\text{foci:} \, \left(7, 7\right), \left(7, -1\right)$$
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$$b)\hspace{.2em}\frac{(x - 8)^2}{25}+ \frac{(y - 9)^2}{16}=1$$ $$\text{center:} \, \left(8, 9\right)$$ $$\text{vertices:} \, \left(13, 9\right), \left(3, 9\right)$$ $$\text{covertices:} \, \left(8, 5\right), \left(8, 13\right)$$ $$\text{foci:} \, \left(11, 9\right), \left(5, 9\right)$$
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#5:
Solutions:
$$a)\hspace{.2em}\frac{x^2}{16}+ \frac{(y + 2)^2}{25}=1$$ $$\text{center:} \, \left(0, -2\right)$$ $$\text{vertices:} \, \left(0, 3\right), \left(0, -7\right)$$ $$\text{covertices:} \, \left(-4, -2\right), \left(4, -2\right)$$ $$\text{foci:} \, \left(0, 1\right), \left(0, -5\right)$$
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$$b)\hspace{.2em}\frac{(x - 1)^2}{9}+ \frac{(y - 3)^2}{25}=1$$ $$\text{center:} \, \left(1, 3\right)$$ $$\text{vertices:} \, \left(1, 8\right), \left(1, -2\right)$$ $$\text{covertices:} \, \left(-2, 3\right), \left(4, 3\right)$$ $$\text{foci:} \, \left(1, 7\right), \left(1, -1\right)$$
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