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

Answer

To find the value of \(\lambda\) such that the line \(y = 3x + \lambda\) touches the hyperbola \(9x^2 - 5y^2 = 45\), we can follow these steps:

### Step 1: Rewrite the hyperbola equation
The given hyperbola is:
\[ 9x^2 - 5y^2 = 45 \]
To put it in standard form, divide the entire equation by 45:
\[ \frac{9x^2}{45} - \frac{5y^2}{45} = 1 \]
This simplifies to:
\[ \frac{x^2}{5} - \frac{y^2}{9} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
\[ a^2 = 5 \quad \text{and} \quad b^2 = 9 \]

### Step 3: Write the equation of the tangent line
The equation of the line is given as:
\[ y = 3x + \lambda \]
Comparing this with the general line equation \(y = mx + c\), we have:
\[ m = 3 \quad \text{and} \quad c = \lambda \]

### Step 4: Use the condition for tangency
For the line to be tangent to the hyperbola, the following condition must hold:
\[ c^2 = a^2 m^2 - b^2 \]
Substituting the values we have:
\[ \lambda^2 = 5(3^2) - 9 \]
Calculating \(5(3^2)\):
\[ 5(9) = 45 \]
Now substituting this back into the equation:
\[ \lambda^2 = 45 - 9 \]
This simplifies to:
\[ \lambda^2 = 36 \]

### Step 5: Solve for \(\lambda\)
Taking the square root of both sides gives:
\[ \lambda = \pm 6 \]

### Final Answer
Thus, the values of \(\lambda\) are:
\[ \lambda = 6 \quad \text{or} \quad \lambda = -6 \]

---