Find the standard form of the equation of the ellipse with the given characteristics.
vertices: (-1,-11), (-1,7),minor axis of length 2

A) $\frac { ( x + 1 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 81 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 81 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
E) $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y - 2 ) ^ { 2 } } { 81 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics. ​
​vertices: (-1,-11), (-1,7),minor axis of length 2
​
A) $\frac { ( x + 1 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 81 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 81 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
E) $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y - 2 ) ^ { 2 } } { 81 } = 1$

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the vertices, which is (-1, -2).

The distance between the center and each vertex on the major axis is 9 units (11 units from -11 to -2 or from -2 to 7, divided by 2). This distance is also equal to ae, where a is the length of the semi-major axis and e is the eccentricity.

Thus, we have 9 = ae.

The length of the minor axis is 2, which is also equal to 2b, where b is the length of the semi-minor axis.

Thus, we have 2 = 2b.

We can then use the formula for the eccentricity of an ellipse:

e = √(1 - (b/a)²)

Solving for b, we get:

b/a = √(1-e²)

b = a√(1-e²)

We can substitute the values we have:

2 = 2b

2 = 2a√(1-e²)

1 = a√(1-e²)

a = 1/√(1-e²)

Substituting this into 9 = ae:

9 = e/√(1-e²)

81 = e²/(1-e²)

81(1-e²) = e²

81 - 81e² = e²

81 = 82e²

e² = 81/82

e ≈ 0.9949

Now that we have the center, the lengths of the semi-major and semi-minor axes, and the eccentricity, we can write the standard form equation:

(x + 1)²/1.0048² + (y + 2)²/0.0988² = 1

Multiplying by the denominators and simplifying, we get:

(x + 1)²/1.0096 - (y + 2)²/0.009775 = 1

This matches answer choice C.

Find the Standard Form of the Equation of the Ellipse (x+1)281+(y+2)24=1\frac { ( x + 1 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 181(x+1)2​+4(y+2)2​=1

Find the Standard Form of the Equation of the Ellipse $\frac { ( x + 1 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$