Question

the eccentricity of the conic section below is

a. closer to 0 than 1
b. closer to 1 than 0

Explanation:

Step1: Recall the formula for eccentricity of an ellipse

The formula for the eccentricity \(e\) of an ellipse is \(e=\frac{c}{a}\), where \(c\) is the distance from the center to a focus and \(a\) is the semi - major axis. Also, for an ellipse \(c^{2}=a^{2}-b^{2}\).

Step2: Analyze the shape of the ellipse

If an ellipse is more circular, \(e\) is closer to \(0\). If it is more elongated, \(e\) is closer to \(1\). From the graph, the ellipse is more circular (less elongated). The semi - major axis \(a\) and semi - minor axis \(b\) values: assume \(a = 6.67\) (semi - major axis length). If the ellipse is more circular, \(b\approx a\). Then \(c=\sqrt{a^{2}-b^{2}}\approx0\) (since \(b\approx a\)).

Step3: Calculate the eccentricity

Using \(e = \frac{c}{a}\), when \(c\approx0\) and \(a>0\), \(e\approx0\)

Answer:

A. closer to 0 than 1