Portion of asymptote of hypebola `(x^(2))/(a^(2)) - (y ^(2))/(b ^(2)) =1` (between centre and the tangent at vertex) in the first quadrant is cut by the `y + lamda (x -b) = 0 (lamda` is a parameter then

Options

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

Answer

To solve the problem step by step, we will analyze the hyperbola, its asymptotes, and the line given in the question.

### Step 1: Identify the asymptotes of the hyperbola
The equation of the hyperbola is given as:

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

The asymptotes of this hyperbola can be derived from the equation by setting the right-hand side to zero:

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

This leads to:

\[ \frac{x^2}{a^2} = \frac{y^2}{b^2} \implies y = \pm \frac{b}{a} x \]

Thus, the asymptotes are:

\[ y = \frac{b}{a} x \quad \text{and} \quad y = -\frac{b}{a} x \]

### Step 2: Identify the tangent at the vertex
The vertices of the hyperbola occur at points \((\pm a, 0)\). The tangent at the vertex \((a, 0)\) can be found using the formula for the tangent to the hyperbola at the vertex:

\[ y = 0 \]

### Step 3: Write the equation of the line
The line given in the problem is:

\[ y + \lambda (x - b) = 0 \]

Rearranging this gives:

\[ y = -\lambda (x - b) = -\lambda x + \lambda b \]

### Step 4: Find the slope of the line
The slope \(m\) of the line is given by the coefficient of \(x\):

\[ m = -\lambda \]

### Step 5: Determine the conditions for the line to intersect the asymptote in the first quadrant
For the line to intersect the asymptote \(y = \frac{b}{a} x\) in the first quadrant, the slope must be negative (since the asymptote in the first quadrant is positively sloped). Therefore, we require:

\[ -\lambda < 0 \implies \lambda > 0 \]

### Step 6: Determine the range of \(\lambda\)
Since \(\lambda\) is a parameter and must be positive, we can conclude:

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

### Final Answer
Thus, the portion of the asymptote of the hyperbola that is cut by the line in the first quadrant is valid for:

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