In the context of conic sections, a “completely flat ellipse” corresponds to a degenerate case where the minor axis shrinks to 0 (so $b=0$ while $a>0$). Using the ellipse eccentricity formula $e=\frac{c}{a}$ with $c^2=a^2-b^2$, what is the eccentricity of this completely flat ellipse?

• A. $e=0$
• B. $e=2$
• C. $e=0.5$
• D. $e=1$

Answer: (D)

Verified step by step guidance

1. Recall the formula for the eccentricity of an ellipse: \(e = \frac{c}{a}\), where \(a\) is the semi-major axis and \(c\) is the distance from the center to a focus.
2. Use the relationship between \(a\), \(b\), and \(c\): \(c^2 = a^2 - b^2\). This relates the semi-major axis \(a\), the semi-minor axis \(b\), and the focal distance \(c\).
3. Since the ellipse is completely flat, the minor axis shrinks to zero, so set \(b = 0\) in the equation \(c^2 = a^2 - b^2\) to get \(c^2 = a^2\).
4. Take the square root of both sides to find \(c = a\) (considering \(a > 0\)).
5. Substitute \(c = a\) back into the eccentricity formula \(e = \frac{c}{a}\) to find \(e = \frac{a}{a} = 1\), which means the eccentricity of a completely flat ellipse is 1.