Question

An ellipse has a center at the origin, a vertex along the major axis at (10, 0), and a focus at (8, 0). Which equation represents this ellipse?

Answer 1

Answer: A

Step-by-step explanation:

Answer 2

Check the picture below.  So the ellipse looks more or less like so.

since the major axis is over the x-axis, is a horizontal ellipse, notice the "a" component length and the value for "c".

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

$\bf \begin{cases} h=0\\ k=0\\ a=10\\ c=8 \end{cases}\implies 8=\sqrt{a^2-b^2}\implies 8^2=a^2-b^2\implies b^2=10^2-8^2 \\\\\\ b=\sqrt{100-64}\implies b=6 \\\\\\ \cfrac{(x-0)^2}{10^2}+\cfrac{(y-0)^2}{6^2}=1\implies \cfrac{x^2}{100}+\cfrac{y^2}{36}=1$