Question

An elliptical mirror measures 12 inches wide and 10 inches high. The ellipse is centered at (0, 48) on a coordinate plane, where units are in inches. Which equation represents the mirror?

Answer 1

The major axis measures 12 in, and the minor axis measures 10 in. The equation of an ellipse centered at (h, k) is given by the expression:

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

Where a is the measure of the semi-major(minor) axis, and b is the measure of the semi-minor(major) axis. In this case, the semi-major axis is horizontal (because it is 12 inches wide), so:

$\begin{gathered} a=\frac{12}{2}=6 \\ b=\frac{10}{2}=5 \end{gathered}$

Now, if the center is at (0, 48), then h = 0 and k = 48. Using these values on the equation of the ellipse:

$\begin{gathered} \frac{(x-0)^2}{6^2}+\frac{(y-48)^2}{5^2}=1 \\ \Rightarrow\frac{x^2}{36}+\frac{(y-48)^2}{25}=1 \end{gathered}$