A hyperbola passes through (3, 3) and the length of its conjugate axis is 8. Find the length of the latus rectum.

Answer

To solve the problem step by step, we will follow the same reasoning as in the video transcript.

### Step 1: Understand the equation of the hyperbola
The standard form of the equation of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \(2b\) is the length of the conjugate axis and \(2a\) is the length of the transverse axis.

### Step 2: Use the information given
We know that the hyperbola passes through the point (3, 3) and the length of the conjugate axis is 8. Therefore, we can set up our equations.

1. Since the length of the conjugate axis is 8, we have:
\[ 2b = 8 \implies b = 4 \]
Thus, \(b^2 = 16\).

### Step 3: Substitute the point into the hyperbola equation
Substituting the point (3, 3) into the hyperbola equation:
\[ \frac{3^2}{a^2} - \frac{3^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{9}{a^2} - \frac{9}{16} = 1 \]

### Step 4: Solve for \(a^2\)
Rearranging the equation gives:
\[ \frac{9}{a^2} = 1 + \frac{9}{16} \]
Finding a common denominator for the right side:
\[ 1 = \frac{16}{16} \implies 1 + \frac{9}{16} = \frac{16 + 9}{16} = \frac{25}{16} \]
Thus, we have:
\[ \frac{9}{a^2} = \frac{25}{16} \]
Cross-multiplying gives:
\[ 9 \cdot 16 = 25 \cdot a^2 \implies 144 = 25a^2 \]
Now, solving for \(a^2\):
\[ a^2 = \frac{144}{25} \]

### Step 5: Find \(a\)
Taking the square root:
\[ a = \frac{12}{5} \]

### Step 6: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of a hyperbola is given by the formula:
\[ L = \frac{2a^2}{b} \]
Substituting the values we found:
\[ L = \frac{2 \cdot \frac{144}{25}}{4} = \frac{288}{25 \cdot 4} = \frac{288}{100} = \frac{72}{25} \]

### Final Answer
The length of the latus rectum is:
\[ \frac{72}{25} \]