If `(4,0) and (-4,0)` be the vertices and `(6,0) and (-6, 0)` be the foci of a hyperbola, then its eccentricity is

Options

A. `5/2`
B. `2`
C. `3/2`
D. `sqrt2`

Answer

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

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

The foci of the hyperbola are given as \( (6, 0) \) and \( (-6, 0) \). The distance from the center to each focus is \( c \). Therefore, we have:
\[ c = 6 \]

### Step 2: Use the relationship between \( a \), \( b \), and \( c \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by the equation:
\[ c^2 = a^2 + b^2 \]

### Step 3: Calculate \( b^2 \)
Substituting the known values of \( a \) and \( c \) into the equation:
\[ 6^2 = 4^2 + b^2 \]
\[ 36 = 16 + b^2 \]
\[ b^2 = 36 - 16 = 20 \]

### Step 4: Calculate the eccentricity \( e \)
The eccentricity \( e \) of a hyperbola is defined as:
\[ e = \frac{c}{a} \]
Substituting the values of \( c \) and \( a \):
\[ e = \frac{6}{4} = \frac{3}{2} \]

### Conclusion
The eccentricity of the hyperbola is:
\[ \boxed{\frac{3}{2}} \]

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