### 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.

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
Conic Sections: The Ellipse Practice Test:

#1:

Instructions: Sketch the graph, state the vertices, and foci.

$$a)\hspace{.2em}\frac{x^2}{36}+ \frac{(y + 2)^2}{4}=1$$

$$b)\hspace{.2em}\frac{(x + 1)^2}{16}+ \frac{(y - 1)^2}{25}=1$$

#2:

Instructions: Sketch the graph, state the vertices, and foci.

$$a)\hspace{.2em}16x^2 + 49y^2 + 98y - 735=0$$

$$b)\hspace{.2em}36x^2 + 16y^2 - 36x - 16y - 563=0$$

#3:

Instructions: Write in standard form.

$$a)\hspace{.2em}\text{vertices}:(-3, -10), (-13, -10)$$ $$\text{foci}:(-4, -10), (-12, -10)$$

$$b)\hspace{.2em}\text{vertices}:(-2, 23), (-2, -3)$$ $$\text{foci}:(-2, 15), (-2, 5)$$

#4:

Instructions: Write in standard form.

$$a)\hspace{.2em}\text{vertices}:(7, 8), (7, -2)$$ $$\text{foci}:(7, 7), (7, -1)$$

$$b)\hspace{.2em}\text{vertices}:(13, 9), (3, 9)$$ $$\text{foci}:(11, 9), (5, 9)$$

#5:

Instructions: Write in standard form.

$$a)\hspace{.2em}\text{vertices}:(0, 3), (0, -7)$$ $$\text{foci}:(0, 1), (0, -5)$$

$$b)\hspace{.2em}\text{vertices}:(1, 8), (1, -2)$$ $$\text{foci}:(1, 7), (1, -1)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\text{vertices}: (6, -2), (-6, -2)$$ $$\text{foci}:(4 \sqrt{2}, -2), (-4 \sqrt{2}, -2)$$

$$b)\hspace{.2em}\text{vertices}: (-1, 6), (-1, -4)$$ $$\text{foci}:(-1, 4), (-1, -2)$$

#2:

Solutions:

$$a)\hspace{.2em}\text{vertices}: (7, -1), (-7, -1)$$ $$\text{foci}:(\sqrt{33}, -1), (-\sqrt{33}, -1)$$

$$b)\hspace{.2em}\text{vertices}: \left(\frac{1}{2}, \frac{13}{2}\right), \left(\frac{1}{2}, -\frac{11}{2}\right)$$ $$\text{foci}:\left(\frac{1}{2}, \frac{4\sqrt{5}+ 1}{2}\right), \left(\frac{1}{2}, \frac{-4 \sqrt{5}+ 1}{2}\right)$$

#3:

Solutions:

$$a)\hspace{.2em}\frac{(x + 8)^2}{25}+ \frac{(y + 10)^2}{9}=1$$

$$b)\hspace{.2em}\frac{(x + 2)^2}{144}+ \frac{(y - 10)^2}{169}=1$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{(x - 7)^2}{9}+ \frac{(y - 3)^2}{25}=1$$

$$b)\hspace{.2em}\frac{(x - 8)^2}{25}+ \frac{(y - 9)^2}{16}=1$$

#5:

Solutions:

$$a)\hspace{.2em}\frac{x^2}{16}+ \frac{(y + 2)^2}{25}=1$$

$$b)\hspace{.2em}\frac{(x - 1)^2}{9}+ \frac{(y - 3)^2}{25}=1$$