Find an equation for the ellipse described.
-Center at $( 4,5 )$; focus at $( 9,5 )$; vertex at $( 11,5 )$
A) $\frac { ( x - 4 ) ^ { 2 } } { 100 } + \frac { ( y + 5 ) ^ { 2 } } { 9 } = 2$
B) $\frac { ( x + 4 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 24 } = 1$
C) $\frac { ( x + 4 ) ^ { 2 } } { 25 } - \frac { ( y - 5 ) ^ { 2 } } { 19 } = 1$
D) $\frac { ( x - 4 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 24 } = 1$

Question

Quizplus · Accepted Answer

The center of the ellipse is at $(4,5)$, indicating that the ellipse equation should be in the form $\frac{(x-4)^2}{a^2} + \frac{(y-5)^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes lengths, respectively. Since the vertex is at $(11,5)$, the distance from the center to the vertex is $11-4=7$, which is the length of the semi-major axis, $a$. Therefore, $a^2=49$. The focus being at $(9,5)$ indicates that the ellipse is horizontal, and the distance from the center to the focus (c) is $9-4=5$. The relationship between $a$, $b$, and $c$ in an ellipse is $a^2=b^2+c^2$. However, without the direct value of $b^2$, we can't accurately determine $b^2$ from the given information alone, but we know the correct equation must have $\frac{(x-4)^2}{49}$ as part of it, making option D the correct choice by process of elimination and the correct form of the equation.

Find an Equation for the Ellipse Described (4,5)( 4,5 )(4,5) ; Focus At

Find an Equation for the Ellipse Described $( 4,5 )$ ; Focus At