Equations of Ellipses Solutions

• Recall what you already know about ellipses:

• If you are given the vertices and co-vertices, you must also determine the distance of the major and minor axes to calculate the    value of a and b    . 

• The    midpoint     of the major or minor axis is the center of the ellipse. 

• The    endpoints     (vertices and co-vertices) determine the domain and range. 

  •    Domain     is between $h–a$ and $h+a$.

  •    Range     is between $k–b$ and $k+b$.

• For an ellipse equation that is not in standard form, rewrite it using this method:

• •    Complete the square     for both x and y.

• Then divide all terms by the value of the    constant     so that the equation equals 1.

Example 5

Write the equation in standard form. Name the domain and range in set builder notation.

$9x^2+4y^2+18x–16y=11$

$$\begin{gathered} 9x^2+18x+4y^2–16y=11 \\ 9\left(x^2+2x+\boxed{{\left(\frac{2}{2}\right)}^2} \right)+4\left(y^2–4y+\boxed{{\left(\frac{–4}{2}\right)}^2} \right)=11+9\boxed{{\left(\frac{2}{2}\right)}^2} +4\boxed{{\left(\frac{–4}{2}\right)}^2} \\ 9{\left(x+1\right)}^2+4{\left(y–2\right)}^2=11+9\left(1\right)+4\left(4\right) \\ \frac{9{\left(x+1\right)}^2}{36}+\frac{4{\left(y–2\right)}^2}{36}=\frac{36}{36} \end{gathered}$$

$$\begin{gathered} \frac{{\left(x+1\right)}^2}{4}+\frac{{\left(y–2\right)}^2}{9}=1 \\ \mathrm{domain} \left\{x|x\in \mathrm{ℝ}, –3\le x\le 1\right\} \\ \mathrm{range} \left\{y|y\in \mathrm{ℝ}, –1\le y\le 5\right\} \end{gathered}$$

Example 6

Write the equation of an ellipse with endpoints: $\left(–17, 0\right)\mathbf{,}\mathbf{ }\left(8, \sqrt{13}\right)\mathbf{,}\mathbf{ }\left(8, –\sqrt{13}\right)\mathbf{,}\mathbf{ }\left(33, 0\right)\mathbf{ }$

$\begin{array}{ccc}\left(–17, 0\right), \left(33, 0\right)& & \left(8, \sqrt{13}\right), \left(8, –\sqrt{13}\right)\\ \left(\frac{–17+33}{2}, 0\right)& & \left(\frac{8+8}{2},\frac{\sqrt{13}+–\sqrt{13} }{2}\right)\\ \left(8, 0\right)& & \left(8, 0\right)\end{array}$

$\frac{{\left(x–8\right)}^2}{625}+\frac{y^2}{13}=1$

Example 7

Write the equation of an ellipse tangent to $\mathit{x}\mathbf{=}\mathbf{–}\mathbf{4}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{=}\mathbf{–}\mathbf{5}$, and the x- and y-axis.