What is the equation of the hyperbola having latus rectum and eccentricity 8 and `(3)/(sqrt(5))` respectively?

Options

A. `(x^(2))/(25)-(y^(2))/(20)=1`
B. `(x^(2))/(40)-(y^(2))/(20)=1`
C. `(x^(2))/(40)-(y^(2))/(30)=1`
D. `(x^(2))/(30) -(y^(2))/(25)=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 relationships
The latus rectum \( L \) of a hyperbola is given by the formula:
\[ L = \frac{b^2}{a} \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

The eccentricity \( e \) of a hyperbola is given by:
\[ e = \frac{c}{a} \]
where \( c \) is the distance from the center to the foci, and it is related to \( a \) and \( b \) by:
\[ c^2 = a^2 + b^2 \]

### Step 2: Substitute the known values
From the problem, we have:
- Latus rectum \( L = 8 \)
- Eccentricity \( e = \frac{3}{\sqrt{5}} \)

Using the latus rectum formula:
\[ 8 = \frac{b^2}{a} \]
This implies:
\[ b^2 = 8a \quad \text{(1)} \]

Using the eccentricity formula:
\[ \frac{3}{\sqrt{5}} = \frac{c}{a} \implies c = \frac{3a}{\sqrt{5}} \quad \text{(2)} \]

### Step 3: Find \( c^2 \)
Substituting equation (2) into the relationship \( c^2 = a^2 + b^2 \):
\[ \left(\frac{3a}{\sqrt{5}}\right)^2 = a^2 + b^2 \]
This simplifies to:
\[ \frac{9a^2}{5} = a^2 + b^2 \]
Substituting \( b^2 \) from equation (1):
\[ \frac{9a^2}{5} = a^2 + 8a \]

### Step 4: Simplify the equation
Rearranging gives:
\[ \frac{9a^2}{5} - a^2 - 8a = 0 \]
Multiplying through by 5 to eliminate the fraction:
\[ 9a^2 - 5a^2 - 40a = 0 \]
This simplifies to:
\[ 4a^2 - 40a = 0 \]
Factoring out \( 4a \):
\[ 4a(a - 10) = 0 \]
Thus, \( a = 0 \) or \( a = 10 \). Since \( a \) cannot be zero, we have:
\[ a = 10 \]

### Step 5: Find \( b^2 \)
Using \( a = 10 \) in equation (1):
\[ b^2 = 8a = 8 \times 10 = 80 \]

### Step 6: 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 \( a^2 = 100 \) and \( b^2 = 80 \):
\[ \frac{x^2}{100} - \frac{y^2}{80} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{x^2}{100} - \frac{y^2}{80} = 1 \]