If the line `y=3x+lambda` touches the hyperbola `9x^(2)-5y^(2)=45`, then the value of `lambda` is

Options

A. 36
B. 45
C. 6
D. 15

Answer

To solve the problem, we need to determine the value of \(\lambda\) such that the line \(y = 3x + \lambda\) is a tangent to the hyperbola given by the equation \(9x^2 - 5y^2 = 45\).

### Step-by-Step Solution:

1. **Convert the Hyperbola to Standard Form**:
The given hyperbola is:
\[ 9x^2 - 5y^2 = 45 \]
To convert this to standard form, divide the entire equation by 45:
\[ \frac{9x^2}{45} - \frac{5y^2}{45} = 1 \]
Simplifying this gives:
\[ \frac{x^2}{5} - \frac{y^2}{9} = 1 \]
Thus, we have \(a^2 = 5\) and \(b^2 = 9\).

2. **Equation of the Tangent Line**:
The general equation of the tangent to the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) at a point with slope \(m\) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Here, \(m\) is the slope of the tangent line.

3. **Identify the Slope**:
From the equation of the line \(y = 3x + \lambda\), we can see that the slope \(m = 3\).

4. **Substitute Values into the Tangent Equation**:
Now, substituting \(m = 3\), \(a^2 = 5\), and \(b^2 = 9\) into the tangent equation:
\[ y = 3x \pm \sqrt{5 \cdot 3^2 - 9} \]
Calculate \(5 \cdot 3^2 - 9\):
\[ 5 \cdot 9 - 9 = 45 - 9 = 36 \]
Thus, we have:
\[ y = 3x \pm \sqrt{36} \]
Which simplifies to:
\[ y = 3x \pm 6 \]

5. **Compare with the Given Line**:
The equation of the tangent can be expressed as:
\[ y = 3x + 6 \quad \text{or} \quad y = 3x - 6 \]
Comparing this with \(y = 3x + \lambda\), we find:
\[ \lambda = 6 \quad \text{or} \quad \lambda = -6 \]

6. **Final Answer**:
Since the problem asks for the value of \(\lambda\) and typically we take the positive value, we conclude:
\[ \lambda = 6 \]