Two tangents are drawn from a point `(-4, 3)` to the parabola `y^(2)=16x`. If `alpha` is the angle between them, then the value of `cos alpha` is

Options

A. 0
B. `(1)/(2)`
C. `(sqrt3)/(2)`
D. `(1)/(sqrt2)`

Answer

To find the value of \( \cos \alpha \) where \( \alpha \) is the angle between the two tangents drawn from the point \((-4, 3)\) to the parabola \( y^2 = 16x \), we can follow these steps:

### Step 1: Write the equation of the parabola
The given parabola is:
\[ y^2 = 16x \]

### Step 2: Find the equation of the tangent to the parabola
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point \((at^2, 2at)\) is given by:
\[ y = mx + \frac{a}{m} \]
Here, \( a = 4 \) (since \( 16 = 4a \)), so the equation of the tangent becomes:
\[ y = mx + \frac{4}{m} \]

### Step 3: Substitute the point \((-4, 3)\) into the tangent equation
Since the tangents pass through the point \((-4, 3)\), we substitute \( x = -4 \) and \( y = 3 \) into the tangent equation:
\[ 3 = m(-4) + \frac{4}{m} \]
This simplifies to:
\[ 3 = -4m + \frac{4}{m} \]

### Step 4: Multiply through by \( m \) to eliminate the fraction
Multiplying through by \( m \) gives:
\[ 3m = -4m^2 + 4 \]
Rearranging this, we get:
\[ 4m^2 + 3m - 4 = 0 \]

### Step 5: Solve the quadratic equation for \( m \)
Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4, b = 3, c = -4 \):
\[ m = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 4 \cdot (-4)}}{2 \cdot 4} \]
\[ m = \frac{-3 \pm \sqrt{9 + 64}}{8} \]
\[ m = \frac{-3 \pm \sqrt{73}}{8} \]

Let the slopes of the tangents be \( m_1 \) and \( m_2 \):
\[ m_1 = \frac{-3 + \sqrt{73}}{8}, \quad m_2 = \frac{-3 - \sqrt{73}}{8} \]

### Step 6: Calculate the angle between the tangents
The angle \( \alpha \) between the two tangents can be calculated using the formula:
\[ \tan \alpha = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
First, we find \( m_1 + m_2 \) and \( m_1 m_2 \):
- Sum of slopes \( m_1 + m_2 = -\frac{3}{4} \)
- Product of slopes \( m_1 m_2 = -1 \)

Since \( m_1 m_2 = -1 \), the tangents are perpendicular:
\[ \tan \alpha = \infty \implies \alpha = \frac{\pi}{2} \]

### Step 7: Find \( \cos \alpha \)
Since \( \alpha = \frac{\pi}{2} \):
\[ \cos \alpha = \cos \left( \frac{\pi}{2} \right) = 0 \]

Thus, the value of \( \cos \alpha \) is:
\[ \boxed{0} \]