Two tangents are drawn from the point `(-2, - 1)` to the parabola `y^2 = 4x` . If a is the angle between these tangents, then `alpha` =

Options

A. 3
B. `1//3`
C. 2
D. `1//2`

Answer

To find the angle \( \alpha \) between the two tangents drawn from the point \((-2, -1)\) to the parabola \(y^2 = 4x\), we can follow these steps:

### Step 1: Identify the parabola and the point
The equation of the parabola is given as \(y^2 = 4x\). The point from which the tangents are drawn is \(P(-2, -1)\).

### Step 2: Write the equation of the tangent
The general equation of the tangent to the parabola \(y^2 = 4x\) at a point with slope \(m\) is given by:
\[ y = mx + \frac{1}{m} \]
Since the tangent must pass through the point \((-2, -1)\), we substitute these coordinates into the tangent equation:
\[ -1 = -2m + \frac{1}{m} \]

### Step 3: Rearranging the equation
To eliminate the fraction, multiply through by \(m\):
\[ -m = -2m^2 + 1 \]
Rearranging gives us:
\[ 2m^2 - m - 1 = 0 \]

### Step 4: Solve the quadratic equation
Now, we can solve the quadratic equation \(2m^2 - m - 1 = 0\) using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 2\), \(b = -1\), and \(c = -1\):
\[ m = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ m = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm 3}{4} \]
Thus, the roots are:
\[ m_1 = 1 \quad \text{and} \quad m_2 = -\frac{1}{2} \]

### Step 5: Find the difference of slopes
Now, we calculate \(m_1 - m_2\):
\[ m_1 - m_2 = 1 - \left(-\frac{1}{2}\right) = 1 + \frac{1}{2} = \frac{3}{2} \]

### Step 6: Find the product of slopes
Next, we calculate \(m_1 \cdot m_2\):
\[ m_1 \cdot m_2 = 1 \cdot \left(-\frac{1}{2}\right) = -\frac{1}{2} \]

### Step 7: Use the formula for the angle between two lines
The formula for the angle \( \alpha \) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \alpha = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting the values we found:
\[ \tan \alpha = \frac{\frac{3}{2}}{1 - \frac{1}{2}} = \frac{\frac{3}{2}}{\frac{1}{2}} = 3 \]

### Step 8: Find the angle \( \alpha \)
Thus, we have:
\[ \alpha = \tan^{-1}(3) \]

### Final Answer
Therefore, the value of \( \alpha \) is:
\[ \alpha = \tan^{-1}(3) \]