Question

The equation of an ellipse is given by (x-3)^2/25 + (y-6)^2/4=1

(a) Identify the coordinates of the center of the ellipse.
(b) Find the length of the major and minor axes.
(c) Find the coordinates of the foci.
(d) Graph the ellipse. Label the center and foci.

Answer 1

`\bf \cfrac{(x-3)^2}{25}+\cfrac{(y-6)^2}{4}=1\iff\cfrac{(x-3)^2}{5^2}+\cfrac{(y-6)^2}{2^2}=1\\\\ -----------------------------\\\\ \textit{ellipse, horizontal major axis}\\\\ \cfrac{(x-{{ h}})^2}{{{ a}}^2}+\cfrac{(y-{{ k}})^2}{{{ b}}^2}=1 \qquad \begin{cases} center\ ({{ h}},{{ k}})\\ vertices\ ({{ h}}\pm a, {{ k}})\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{{{ a }}^2-{{ b }}^2}\\ foci=h\ ,\ h\pm c \end{cases}`


now, for an ellipse, the major axis is on the axis with the variable with the larger denominator, in this case, the larger denominator is under the "x", thus the major axis is over the x-axis

and surely you can tell what the major and minor axis are, a+a and b+b

and the foci are one at h, h+c and the other at h, h-c

and the ellipse looks more or less like the picture below