Find the standard form of the equation of the ellipse with the following characteristics.
foci: $( \pm 4,0 )$
major axis of length: 14
A) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$
B) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 16 } = 1$
E) $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 180 } = 1$

Question

Quizplus · Accepted Answer

The major axis is horizontal and the foci are on the x-axis. The distance between the foci is 2c = 8, so c = 4. The length of the major axis is 2a = 14, so a = 7. The standard form of the equation of an ellipse with a horizontal major axis is $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$. Substituting the values of a and c, we get $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. To find the value of b, we use the equation $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$. Substituting the values of a and c, we get $4 ^ { 2 } = 7 ^ { 2 } - b ^ { 2 }$. Solving for b, we get $b ^ { 2 } = 33$. Substituting this value of b into the equation of the ellipse, we get $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$.

Find the Standard Form of the Equation of the Ellipse (±4,0)( \pm 4,0 )(±4,0)

Find the Standard Form of the Equation of the Ellipse $( \pm 4,0 )$