Find the equation of the ellipse satisfying the given conditions.
-(0, 6), (0, -6), (3, 0), (-3, 0) are the endpoints of the major and minor axes.

Question

Quizplus · Accepted Answer

Based on the given endpoints of the major and minor axes, we can determine that the center of the ellipse is at the origin (0,0), and the length of the major and minor axes are 6 and 3, respectively. Thus, the equation of the ellipse is 
$$\frac{x^2}{3^2}+\frac{y^2}{6^2}=1$$
which simplifies to 
$$\frac{x^2}{9}+\frac{y^2}{36}=1$$
The coefficients in choice C match this equation, so it is the correct answer. The standard form of the equation of an ellipse with center at the origin is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

From the given endpoints of the major and minor axes, we can see that the length of the major axis is $2a=6-(-6)=12$ and the length of the minor axis is $2b=3-(-3)=6$. Therefore, $a=6$ and $b=3$.

Since the center of the ellipse is at the origin, the equation simplifies to:

$\frac{x^2}{36}+\frac{y^2}{9}=1$

This matches choice C.

Find the Equation of the Ellipse Satisfying the Given Conditions