Find the standard form of the equation of the ellipse with the given characteristics.
Center: $( 15,5 )$ ; a = 3b; Foci: $( 5,5 ) , ( 25,5 )$ , where a - thr semimajor axis, b - semiminor axis.

A) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
B) $\frac { x ^ { 2 } } { 225 } - \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 200 } = 1$
D) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics. 
Center: $( 15,5 )$ ; a = 3b; Foci: $( 5,5 ) , ( 25,5 )$ , where a - thr semimajor axis, b - semiminor axis.

A) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
B) $\frac { x ^ { 2 } } { 225 } - \frac { ( y - 5 ) ^ { 2 } } { 200 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 200 } = 1$
D) $\frac { ( x - 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 225 } + \frac { ( y + 5 ) ^ { 2 } } { 200 } = 1$

Quizplus · Accepted Answer

The standard form of the equation of an ellipse centered at $(h, k)$ is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ for a horizontal ellipse, and $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ for a vertical ellipse. Given that the center is $(15,5)$, the foci are $(5,5)$ and $(25,5)$, indicating a horizontal ellipse. The distance between the center and either focus is 10 units, which is the value of $c$ (where $c^2 = a^2 - b^2$). Since $a = 3b$, we can find $a$ and $b$ by solving the system of equations derived from these conditions. The correct equation must have the center at $(15,5)$, which is mistakenly not reflected in any of the choices, indicating a potential misunderstanding in the transcription of the options or the interpretation of the ellipse's parameters. However, assuming the intention was to have the center at $(15,5)$ and considering the given options, none directly match the expected standard form equation based on the provided center and the relationship between $a$ and $b$. The correct standard form equation would involve terms like $\frac{(x-15)^2}{a^2}$ and $\frac{(y-5)^2}{b^2}$, with $a$ and $b$ calculated from the given conditions. The provided options do not accurately reflect these calculations, suggesting a need for reevaluation of the answer choices or the question's details.

Find the Standard Form of the Equation of the Ellipse (15,5)( 15,5 )(15,5)

Find the Standard Form of the Equation of the Ellipse $( 15,5 )$