A hyperbola passes through (3,2) and (-17, 12) and has its centre at origin and transverse axis along X-axis . Then length of its transverse axis is

Options

A. 2
B. 4
C. 6
D. none of these

Answer

To find the length of the transverse axis of the hyperbola that passes through the points (3, 2) and (-17, 12) with its center at the origin and transverse axis along the x-axis, we can follow these steps:

### Step 1: Write the equation of the hyperbola
Since the hyperbola has its center at the origin and the transverse axis along the x-axis, its equation is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 2: Substitute the first point (3, 2) into the equation
Substituting the point (3, 2) into the hyperbola equation:
\[ \frac{3^2}{a^2} - \frac{2^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{9}{a^2} - \frac{4}{b^2} = 1 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point (-17, 12) into the equation
Now, substitute the point (-17, 12) into the hyperbola equation:
\[ \frac{(-17)^2}{a^2} - \frac{12^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{289}{a^2} - \frac{144}{b^2} = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \(\frac{9}{a^2} - \frac{4}{b^2} = 1\)
2. \(\frac{289}{a^2} - \frac{144}{b^2} = 1\)

To eliminate \(b^2\), we can multiply the first equation by a suitable factor. Let's multiply Equation 1 by 36:
\[ 36\left(\frac{9}{a^2} - \frac{4}{b^2}\right) = 36 \implies \frac{324}{a^2} - \frac{144}{b^2} = 36 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 3
Now, subtract Equation 2 from Equation 3:
\[ \left(\frac{324}{a^2} - \frac{144}{b^2}\right) - \left(\frac{289}{a^2} - \frac{144}{b^2}\right) = 36 - 1 \]
This simplifies to:
\[ \frac{324 - 289}{a^2} = 35 \]
\[ \frac{35}{a^2} = 35 \]
Thus, we find:
\[ a^2 = 1 \]

### Step 6: Find the length of the transverse axis
The length of the transverse axis is given by \(2a\). Since \(a^2 = 1\), we have:
\[ a = 1 \]
Therefore, the length of the transverse axis is:
\[ 2a = 2 \times 1 = 2 \]

### Final Answer
The length of the transverse axis is \(2\).

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