Question

If the foci and vertices of an ellipse are respectively $(\pm 2,0)$ and $(\pm 3,0)$ then its eccentricity is

• A. $\frac{2}{3}$
• B. $\frac{\sqrt{5}}{3}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{1}{2}$
• E. $\frac{1}{\sqrt{2}}$

Hint:

Logic Tip: The eccentricity of an ellipse is simply the ratio of the distance to the focus over the distance to the vertex: $e = \frac{ae}{a}$. Reading the coordinates directly gives you the answer instantly: $2 / 3$.

Answer 1

The Correct Option is A

Concept:
For a standard ellipse centered at the origin with its major axis along the x-axis, the coordinates of the vertices are $(\pm a, 0)$ and the coordinates of the foci are $(\pm ae, 0)$, where $a$ is the semi-major axis and $e$ is the eccentricity.
Step 1: Identify the values from the given coordinates.
The vertices are given as $(\pm 3, 0)$. By comparing this to the standard form $(\pm a, 0)$, we get: $$a = 3$$ The foci are given as $(\pm 2, 0)$. By comparing this to the standard form $(\pm ae, 0)$, we get: $$ae = 2$$
Step 2: Solve for the eccentricity (e).
Substitute the value of $a = 3$ into the foci equation: $$(3)e = 2$$ $$e = \frac{2}{3}$$