If the line `y = mx + sqrt(a^(2) m^(2) -b^(2)), m = (1)/(2)` touches the hyperbola `(x^(2))/(16)-(y^(2))/(3) =1` at the point `(4 sec theta, sqrt(3) tan theta)` then `theta` is

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(2pi)/(3)`
D. `(pi)/(6)`

Answer

To solve the problem step by step, we need to find the value of \( \theta \) given that the line \( y = mx + \sqrt{a^2 m^2 - b^2} \) touches the hyperbola \( \frac{x^2}{16} - \frac{y^2}{3} = 1 \) at the point \( (4 \sec \theta, \sqrt{3} \tan \theta) \) with \( m = \frac{1}{2} \).

### Step 1: Find the derivative of the hyperbola
The equation of the hyperbola is given by:
\[ \frac{x^2}{16} - \frac{y^2}{3} = 1 \]
To find the slope of the tangent line at any point on the hyperbola, we differentiate implicitly:
\[ \frac{d}{dx}\left(\frac{x^2}{16}\right) - \frac{d}{dx}\left(\frac{y^2}{3}\right) = 0 \]
This gives:
\[ \frac{2x}{16} - \frac{2y}{3} \frac{dy}{dx} = 0 \]
Rearranging, we find:
\[ \frac{2y}{3} \frac{dy}{dx} = \frac{2x}{16} \]
Thus,
\[ \frac{dy}{dx} = \frac{3x}{16y} \]

### Step 2: Substitute the point into the derivative
We substitute the point \( (4 \sec \theta, \sqrt{3} \tan \theta) \) into the derivative:
\[ \frac{dy}{dx} = \frac{3(4 \sec \theta)}{16(\sqrt{3} \tan \theta)} = \frac{12 \sec \theta}{16 \sqrt{3} \tan \theta} = \frac{3 \sec \theta}{4 \sqrt{3} \tan \theta} \]

### Step 3: Set the slope equal to \( m \)
Since the line touches the hyperbola, the slope of the tangent line must equal \( m \):
\[ \frac{3 \sec \theta}{4 \sqrt{3} \tan \theta} = \frac{1}{2} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ 3 \sec \theta = 2 \cdot 4 \sqrt{3} \tan \theta \]
This simplifies to:
\[ 3 \sec \theta = 8 \sqrt{3} \tan \theta \]

### Step 5: Substitute \( \sec \theta \) and \( \tan \theta \)
Recall that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ 3 \cdot \frac{1}{\cos \theta} = 8 \sqrt{3} \cdot \frac{\sin \theta}{\cos \theta} \]
Cancelling \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ 3 = 8 \sqrt{3} \sin \theta \]

### Step 6: Solve for \( \sin \theta \)
Rearranging gives:
\[ \sin \theta = \frac{3}{8 \sqrt{3}} = \frac{3 \sqrt{3}}{24} = \frac{\sqrt{3}}{8} \]

### Step 7: Find \( \theta \)
To find \( \theta \), we recognize that \( \sin \theta = \frac{\sqrt{3}}{2} \) corresponds to angles of \( 60^\circ \) or \( 120^\circ \) (in the first and second quadrants, respectively).

Thus, the possible values for \( \theta \) are:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = 120^\circ \]

### Final Answer
The value of \( \theta \) is \( 60^\circ \) or \( 120^\circ \).