The hyperbola `(x^(2))/(a^(2)) -(y^(2))/(b^(2))=1` passes through the point `(3sqrt(5),1)` and the length of its latus rectum is `(4)/(3)` units. The length of the conjugate axis is:

Options

A. 2 units
B. 3 units
C. 4 units
D. 5 units

Answer

To solve the problem step by step, we will follow the given information about the hyperbola and derive the necessary values.

### Step 1: Write the equation of the hyperbola
The equation of the hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 2: Substitute the point into the hyperbola equation
The hyperbola passes through the point \((3\sqrt{5}, 1)\). Substituting these values into the hyperbola equation gives:
\[ \frac{(3\sqrt{5})^2}{a^2} - \frac{1^2}{b^2} = 1 \]
Calculating \( (3\sqrt{5})^2 \):
\[ \frac{45}{a^2} - \frac{1}{b^2} = 1 \]
This is our first equation.

### Step 3: Use the length of the latus rectum
The length of the latus rectum of a hyperbola is given by:
\[ \text{Latus Rectum} = \frac{2b^2}{a} \]
According to the problem, the length of the latus rectum is \(\frac{4}{3}\):
\[ \frac{2b^2}{a} = \frac{4}{3} \]
From this, we can express \(b^2\) in terms of \(a\):
\[ 2b^2 = \frac{4a}{3} \implies b^2 = \frac{2a}{3} \]
This is our second equation.

### Step 4: Substitute \(b^2\) into the first equation
Now we substitute \(b^2 = \frac{2a}{3}\) into the first equation:
\[ \frac{45}{a^2} - \frac{1}{\frac{2a}{3}} = 1 \]
This simplifies to:
\[ \frac{45}{a^2} - \frac{3}{2a} = 1 \]
To eliminate the fractions, multiply through by \(2a^2\):
\[ 90 - 3a = 2a^2 \]
Rearranging gives us:
\[ 2a^2 + 3a - 90 = 0 \]

### Step 5: Solve the quadratic equation
Now we can solve the quadratic equation \(2a^2 + 3a - 90 = 0\) using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-90)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ = \frac{-3 \pm \sqrt{9 + 720}}{4} = \frac{-3 \pm \sqrt{729}}{4} = \frac{-3 \pm 27}{4} \]
This gives us two possible values for \(a\):
\[ a = \frac{24}{4} = 6 \quad \text{and} \quad a = \frac{-30}{4} = -7.5 \quad (\text{not valid since } a > 0) \]
Thus, \(a = 6\).

### Step 6: Find \(b^2\)
Now we can find \(b^2\) using \(b^2 = \frac{2a}{3}\):
\[ b^2 = \frac{2 \cdot 6}{3} = 4 \]

### Step 7: Find the length of the conjugate axis
The length of the conjugate axis is given by \(2b\):
\[ b = \sqrt{b^2} = \sqrt{4} = 2 \]
Thus, the length of the conjugate axis is:
\[ 2b = 2 \cdot 2 = 4 \]

### Final Answer
The length of the conjugate axis is \(4\) units.

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