Write the equation of the ellipse that has its center at the origin, focus at $( 2,0 )$ and $\frac { 3 } { 2 }$ is one-half the length of the minor axis.
A) $\quad \frac { 4 x ^ { 2 } } { 25 } + \frac { 4 y ^ { 2 } } { 9 } = 1$
B) $\quad \frac { 4 x ^ { 2 } } { 25 } + \frac { 4 y ^ { 2 } } { 9 } = - 1$
C) $\quad \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$
D) $\quad \frac { 2 x ^ { 2 } } { 3 } + \frac { 2 y ^ { 2 } } { 5 } = 1$
E) $\quad \frac { 4 x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$

Question

Quizplus · Accepted Answer

The equation of an ellipse centered at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. The distance between the center and a focus, $c$, satisfies $c^2 = a^2 - b^2$. Given a focus at $(2,0)$, we have $c = 2$. The minor axis is $2 \times \frac{3}{2} = 3$, so $b = \frac{3}{2}$ and $b^2 = \frac{9}{4}$. Using $c^2 = a^2 - b^2$, we find $a^2 = 4 + \frac{9}{4} = \frac{25}{4}$. Thus, the equation is $\frac{x^2}{(\frac{25}{4})} + \frac{y^2}{(\frac{9}{4})} = 1$, which simplifies to $\frac{4x^2}{25} + \frac{4y^2}{9} = 1$.

Write the Equation of the Ellipse That Has Its Center (2,0)( 2,0 )(2,0)

Write the Equation of the Ellipse That Has Its Center $( 2,0 )$