If the portion of the asymptote between the cnetre and the tangent at the vertex of the hyperbola `(x ^(2))/(a^(2)) - (y ^(2))/(b ^(2)) =1` in the third quadrant is cut by the line `y + lamda (x + a) = 0, lamda `being parameter, then

Options

A. `lamda in R ^(+)`
B. `lamda in R ^(-)`
C. `lamda in (0,1)`
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the hyperbola and its asymptotes
The given hyperbola is

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

The asymptotes of this hyperbola can be derived from the equation by setting it equal to zero:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \]

This simplifies to:

\[ y = \pm \frac{b}{a} x \]

### Step 2: Determine the vertices and tangents
The vertices of the hyperbola are located at \((a, 0)\) and \((-a, 0)\). The tangents at these vertices are given by the equations:

- For the vertex at \((a, 0)\): \(x = a\)
- For the vertex at \((-a, 0)\): \(x = -a\)

### Step 3: Analyze the line given in the problem
The line given in the problem is:

\[ y + \lambda (x + a) = 0 \]

This can be rewritten as:

\[ y = -\lambda x - \lambda a \]

### Step 4: Determine the conditions for intersection in the third quadrant
For the line to intersect the asymptotes in the third quadrant, we need to consider the slopes. The slope of the line is \(-\lambda\).

To ensure that the line intersects the asymptotes in the third quadrant, the slope must be negative. Therefore, we require:

\[ -\lambda < 0 \quad \Rightarrow \quad \lambda > 0 \]

### Step 5: Conclusion
Since we have established that \(\lambda\) must be greater than 0 for the line to intersect the asymptotes in the third quadrant, we conclude that:

\[ \lambda \in (0, \infty) \]