Write the equation of the ellipse that has its center at the origin, focus at $( 1,0 )$ and $\frac { 7 } { 8 }$ is one-half the length of the minor axis.
A) $\quad \frac { 64 x ^ { 2 } } { 113 } + \frac { 64 y ^ { 2 } } { 49 } = 1$
B) $\quad \frac { 64 x ^ { 2 } } { 113 } - \frac { y ^ { 2 } } { 49 } = 1$
C) $\quad \frac { 8 x ^ { 2 } } { 7 } + \frac { 8 y ^ { 2 } } { 15 } = 1$
D) $\frac { x ^ { 2 } } { 113 } + \frac { y ^ { 2 } } { 49 } = 1$
E) $\frac { 64 x ^ { 2 } } { 113 } + \frac { 64 y ^ { 2 } } { 49 } = - 1$

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, $2b$ is the length of the minor axis, and $c$ is the distance from the center to a focus. Given that one focus is at $(1,0)$, we have $c=1$. The length of the minor axis is $2 \times \frac{7}{8} = \frac{7}{4}$, so $b = \frac{7}{4}$ and $b^2 = \frac{49}{16}$. Using $c^2 = a^2 - b^2$, we find $a^2 = c^2 + b^2 = 1 + \frac{49}{16} = \frac{65}{16}$. Thus, the equation is $\frac{x^2}{(\frac{65}{16})} + \frac{y^2}{(\frac{49}{16})} = 1$, which simplifies to $\frac{64x^2}{65} + \frac{64y^2}{49} = 1$, matching closest to option A after correcting the minor calculation mistake in the explanation process.

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 $( 1,0 )$