Write the equation of the ellipse that has its center at the origin, focus at $( 1,0 )$ and $\frac { 3 } { 4 }$ is one-half the length of the minor axis.
A) $\quad \frac { 16 x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$
B) $\quad \frac { 16 x ^ { 2 } } { 25 } + \frac { 16 y ^ { 2 } } { 9 } = - 1$
C) $\quad \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$
D) $\quad \frac { 4 x ^ { 2 } } { 3 } + \frac { 4 y ^ { 2 } } { 7 } = 1$
E) $\quad \frac { 16 x ^ { 2 } } { 25 } + \frac { 16 y ^ { 2 } } { 9 } = 1$

Question

Quizplus · Accepted Answer

The ellipse has its center at the origin, so the equation is of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The focus is at (1,0), so $c = 1$. The relationship $c^2 = a^2 - b^2$ gives $1 = a^2 - b^2$. The minor axis is $2b = \frac{3}{2}$, so $b = \frac{3}{4}$. Solving for $a$ using $b^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$, we find $a^2 = \frac{25}{16}$. Thus, the equation is $\frac{16x^2}{25} + \frac{16y^2}{9} = 1$.

Write the Equation of the Ellipse That Has Its Center (1,0)( 1,0 )(1,0)

Write the Equation of the Ellipse That Has Its Center $( 1,0 )$