The vertices of a hyperbola are ` (2,0 ) ,( -2,0) and` the foci are (3,0) ,(-3,0) .The equation of the hyperbola is

Options

A. `x^(2)/5-y^(2)/4=1`
B. `x^(2)/(4)-y^(2)/5=1`
C. `x^(2)/5-y^(2)/2=1`
D. `x^(2)/2-y^(2)/5=1`

Answer

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

### Step 1: Identify the center of the hyperbola
The vertices of the hyperbola are given as (2, 0) and (-2, 0). The center of the hyperbola is the midpoint of the line segment joining the vertices.

**Calculation:**
\[ \text{Center} = \left( \frac{2 + (-2)}{2}, \frac{0 + 0}{2} \right) = (0, 0) \]

### Step 2: Determine the value of 'a'
The distance from the center to each vertex is 'a'. Since the vertices are at (2, 0) and (-2, 0), we have:

**Calculation:**
\[ a = 2 \]

### Step 3: Identify the foci and determine 'c'
The foci of the hyperbola are given as (3, 0) and (-3, 0). The distance from the center to each focus is 'c'.

**Calculation:**
\[ c = 3 \]

### Step 4: Calculate 'b' using the relationship between a, b, and c
For hyperbolas, the relationship between 'a', 'b', and 'c' is given by the equation:
\[ c^2 = a^2 + b^2 \]

**Calculation:**
\[ c^2 = 3^2 = 9 \quad \text{and} \quad a^2 = 2^2 = 4 \]
\[ 9 = 4 + b^2 \implies b^2 = 9 - 4 = 5 \]

### Step 5: Write the equation of the hyperbola
Since the hyperbola opens horizontally (as the transverse axis is along the x-axis), the standard form of the equation is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

**Substituting the values of a and b:**
\[ \frac{x^2}{2^2} - \frac{y^2}{5} = 1 \implies \frac{x^2}{4} - \frac{y^2}{5} = 1 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]

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