Equation of the hyperbola with latus rectum `(22)/(5)` and eccentricity 6/5 is

Options

A. a) `x^(2)/9-y^(2)/4=1`
B. b) `x^2/(16)-y^(2)/(20)=1`
C. c) `x^(2)/(16)-y^(2)/(9)=1`
D. d) `x^(2)/(25)-y^(2)/(11)=1`

Answer

To find the equation of the hyperbola with a given latus rectum and eccentricity, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Latus rectum \( L = \frac{22}{5} \)
- Eccentricity \( e = \frac{6}{5} \)

### Step 2: Use the Eccentricity Formula
The formula for the eccentricity of a hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the given eccentricity:
\[ \frac{6}{5} = \sqrt{1 + \frac{b^2}{a^2}} \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square root:
\[ \left(\frac{6}{5}\right)^2 = 1 + \frac{b^2}{a^2} \]
This simplifies to:
\[ \frac{36}{25} = 1 + \frac{b^2}{a^2} \]

### Step 4: Rearrange the Equation
Rearranging gives:
\[ \frac{b^2}{a^2} = \frac{36}{25} - 1 \]
Calculating the right-hand side:
\[ \frac{b^2}{a^2} = \frac{36}{25} - \frac{25}{25} = \frac{11}{25} \]

### Step 5: Express \( b^2 \) in Terms of \( a^2 \)
From the equation above, we can express \( b^2 \):
\[ b^2 = \frac{11}{25} a^2 \]

### Step 6: Use the Latus Rectum Formula
The formula for the latus rectum \( L \) of a hyperbola is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the given latus rectum:
\[ \frac{22}{5} = \frac{2b^2}{a} \]

### Step 7: Substitute \( b^2 \) into the Latus Rectum Formula
Substituting \( b^2 = \frac{11}{25} a^2 \):
\[ \frac{22}{5} = \frac{2 \cdot \frac{11}{25} a^2}{a} \]
This simplifies to:
\[ \frac{22}{5} = \frac{22}{25} a \]

### Step 8: Solve for \( a \)
Cross-multiplying gives:
\[ 22 \cdot 25 = 22 \cdot 5 a \]
Dividing both sides by 22:
\[ 25 = 5a \]
Thus:
\[ a = 5 \]

### Step 9: Find \( b \)
Using \( b^2 = \frac{11}{25} a^2 \):
\[ b^2 = \frac{11}{25} \cdot 5^2 = \frac{11}{25} \cdot 25 = 11 \]

### Step 10: Write the Equation of the Hyperbola
The standard form of the hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 = 25 \) and \( b^2 = 11 \):
\[ \frac{x^2}{25} - \frac{y^2}{11} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{x^2}{25} - \frac{y^2}{11} = 1 \]