The eccentricty of an ellipse, the co-ordinates of whose vertices and foci are `(pm 4, 0)` and `(pm sqrt(7), 0)` respectively, is

Options

A. `(sqrt(7))/(4)`
B. `(4)/(sqrt(7))`
C. `(9)/(2)`
D. `(2)/(9)`

Answer

To find the eccentricity of the ellipse given the coordinates of its vertices and foci, we can follow these steps:

### Step 1: Identify the values of \( a \) and \( c \)
The vertices of the ellipse are given as \( (\pm 4, 0) \). This means that the distance from the center to each vertex, which is denoted as \( a \), is 4. Therefore, we have:
\[ a = 4 \]

The foci of the ellipse are given as \( (\pm \sqrt{7}, 0) \). This means that the distance from the center to each focus, denoted as \( c \), is \( \sqrt{7} \). Thus, we have:
\[ c = \sqrt{7} \]

### Step 2: Use the relationship between \( a \), \( b \), and \( c \)
For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by the equation:
\[ c^2 = a^2 - b^2 \]
However, we do not need to find \( b \) to calculate the eccentricity.

### Step 3: Calculate the eccentricity \( e \)
The eccentricity \( e \) of an ellipse is defined as:
\[ e = \frac{c}{a} \]
Substituting the values we found for \( c \) and \( a \):
\[ e = \frac{\sqrt{7}}{4} \]

### Final Answer
Thus, the eccentricity of the ellipse is:
\[ e = \frac{\sqrt{7}}{4} \]

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