Question

Write the equation of a vertical ellipse with a center of (5,-6) with a major axis of 3 and a minor axis of 2

Answer 1

Given:

Center of the ellipse (5,-6)

Major axis = 3

Minor axis = 2

Find-: Equation of ellipse.

Sol:

The general equation of the vertical ellipse is:

$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$

Where,

$\begin{gathered} \text{ Center = }(h,k) \\ \\ \text{ Major axis = 2a} \\ \\ \text{ Minor axis = 2b} \end{gathered}$

So, value of "a" , "b" is:

$\begin{gathered} \text{ Major axis = 3} \\ \\ 2a=3 \\ \\ a=\frac{3}{2} \end{gathered}$

Minor axis is:

$\begin{gathered} 2b=2 \\ \\ b=\frac{2}{2} \\ \\ b=1 \end{gathered}$

So equation become:

$\begin{gathered} \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 \\ \\ \frac{(x-5)^2}{1^2}+\frac{(y-(-6))^2}{(\frac{3}{2})^2}=1 \\ \\ \frac{(x-5)^2}{1}+\frac{(y+6)^2}{\frac{9}{4}}=1 \end{gathered}$

Final equation of ellipse is

$\begin{gathered} \frac{(x-5)^2}{1}+\frac{(y+6)^2}{\frac{9}{4}}=1 \\ \\ (x-5)^2+\frac{4(y+6)^2}{9}=1 \end{gathered}$