Practice 2 Solutions

1. Write the equation of the ellipse with a center $(0, 0)$, a horizontal major axis length of 5, and a minor axis length of 3.

$$\begin{gathered} \mathrm{Center}: \left(0, 0\right) \\ 2a=5 \\ a=2.5 \\ {a}^2=6.25 \\ 2b=3 \\ b=1.5 \\ {b}^2=2.25 \end{gathered}$$

$\frac{x^2}{6.25}+\frac{y^2}{2.25}=1$

2. Write the equation of the ellipse with a center $\left(–23.12, 98.54\right)$, when $a=4\sqrt{5} \mathrm{and} b=2\sqrt{3}$.

$$\begin{gathered} \mathrm{C}\mathrm{enter}: \left(–23.12, 98.54\right) \\ a=4\sqrt{5} \\ {a}^2={\left(4\sqrt{5}\right)}^2=16\sqrt{25}=16·5=80 \\ b=2\sqrt{3} \\ {b}^2={\left(2\sqrt{3}\right)}^2=4\sqrt{9}=4·3=12 \end{gathered}$$

$\frac{{\left(x+23.12\right)}^2}{80}+\frac{{\left(y–98.54\right)}^2}{12}=1$

3. Write the equation of the ellipse with vertices $(6, –4)$ and $(–10, –4)$, and co-vertices $(–2, –1)$ and $(–2, –7)$.

$$\begin{gathered} \mathrm{C}\mathrm{enter}:\left(\frac{6+–10}{2}, \frac{–4+–4 }{2}\right)=\left(–2, –4\right) \\ 2a=16 \\ a=8 \\ {a}^2=64 \\ 2b=6 \\ b=3 \\ {b}^2=9 \end{gathered}$$

$\frac{{\left(x+2\right)}^2}{64}+\frac{{\left(y+4\right)}^2}{9}=1$

4. Write the equation of the ellipse when it is translated 8 units to the left, 4 units up, and the major axis is increased by 2.
   
   $\frac{x^2}{9}+\frac{{(y–3)}^2}{64}=1$

$$\begin{gathered} \mathrm{C}\mathrm{enter}: \left(0–8, 3+4\right)=\left(–8, 7\right) \\ a=3 \\ 2a = 6 \left(\mathrm{minor} \mathrm{axis}\right) \\ b=8 \\ 2b=16 \left(\mathrm{major} \mathrm{axis}\right) \\ \mathrm{N}\mathrm{ew} \mathrm{major} \mathrm{axis}:\hspace{0.17em}16+2=18 \\ b=9 \end{gathered}$$

$\frac{{\left(x+8\right)}^2}{9}+\frac{{\left(y–7\right)}^2}{81}=1$

7. A domed building with an elliptical room measures 23 feet wide and 97 feet long. Write the equation of the footprint of the room centered at the origin.

$$\begin{gathered} \mathrm{C}\mathrm{enter}:\hspace{0.17em}\left(0, 0\right) \\ 2a=23 \\ a=12.5 \\ {a}^2=156.25 \\ 2b=97 \\ b=48.5 \\ {b}^2=2,352.25 \end{gathered}$$

$\frac{x^2}{156.25}+\frac{y^2}{2,352.25}=1$

8. The blueprint for an elliptical stadium shows the horizontal major axis with a length of 4 units and the minor axis length of 2 units. The building will be 20 times the scale of the blueprint. Write the equation of the constructed stadium centered at the origin. 

$$\begin{gathered} 2a=4 \\ a=2 \\ \left(20\right)a=\left(20\right)2=40 \\ 2b=2 \\ b=1 \\ \left(20\right)b=\left(20\right)1=20 \end{gathered}$$

$\frac{x^2}{1600}+\frac{y^2}{400}=1$

9. $100x^2–200x+225y^2–450y=22,175$

$$\begin{gathered} 100\left(x^2–2x\right)+225\left(y^2–2y\right)=22,175 \\ 100\left(x^2–2x+\boxed{{\left(–\frac{2}{2}\right)}^2}\right)+225\left(y^2–2y+\boxed{{\left(–\frac{2}{2}\right)}^2}\right)=22,175+100\boxed{{\left(–\frac{2}{2}\right)}^2}+225\boxed{{\left(–\frac{2}{2}\right)}^2} \\ 100{\left(x–1\right)}^2+225{\left(y–1\right)}^2=22,175+100+225 \\ 100{\left(x–1\right)}^2+225{\left(y–1\right)}^2=22,500 \end{gathered}$$

$\frac{{\left(x–1\right)}^2}{225}+\frac{{\left(y–1\right)}^2}{100}=1$

10. $3x^2+12x+8y^2–48y=12$

$$\begin{gathered} 3\left(x^2+4x\right)+8\left(y^2–6y\right)=12 \\ 3\left(x^2+4x+\boxed{{\left(\frac{4}{2}\right)}^2}\right)+8\left(y^2–6y+\boxed{{\left(–\frac{6}{2}\right)}^2}\right)=12+3\boxed{{\left(\frac{4}{2}\right)}^2}+8\boxed{{\left(–\frac{6}{2}\right)}^2} \\ 3{\left(x+2\right)}^2+8{\left(y–3\right)}^2=12+3\left(4\right)+8\left(9\right) \\ 3{\left(x+2\right)}^2+8\left(y–3\right)=96 \end{gathered}$$

$\frac{{\left(x+2\right)}^2}{32}+\frac{{\left(y–3\right)}^2}{12}=1$

11. Explain how to determine if the ellipse is horizontal or vertical from an equation.

Sample: Determine the values of a and b from the equation. If $a>b$, then the major axis and the ellipse are horizontal. If $b>a$, then the major axis and the ellipse are vertical.

12. Explain how to determine the length of the major and minor axis given the equation of the ellipse.

Sample: In the equation of an ellipse, the values of a and b are squared. To determine the axes, you need to take the square root of a and b. Then double each value to find the length of each axis.