Two tangent are drawn from the point `(-2,-1)` to parabola `y^2=4xdot` if `alpha` is the angle between these tangents, then find the value of `tanalphadot`

Options

A. 3
B. 43468
C. 2
D. 43467

Explanation

To solve the problem of finding the value of \( \tan \alpha \) where \( \alpha \) is the angle between the two tangents drawn from the point \((-2, -1)\) to the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Write the equation of the parabola and the tangent line
The equation of the parabola is given as:
\[ y^2 = 4x \]
For a parabola of the form \( y^2 = 4ax \), the equation of the tangent at a point \((x_1, y_1)\) on the parabola can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent and \( a = 1 \) (since \( 4a = 4 \)).

### Step 2: Substitute the point into the tangent equation
The point from which the tangents are drawn is \((-2, -1)\). Substituting this point into the tangent equation gives:
\[ -1 = m(-2) + \frac{1}{m} \]
This simplifies to:
\[ -1 = -2m + \frac{1}{m} \]

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

### Step 4: Solve the quadratic equation
Now, we can solve the quadratic equation \( 2m^2 - m - 1 = 0 \) using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 2, b = -1, c = -1 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4(2)(-1) = 1 + 8 = 9 \]
Now substituting into the formula:
\[ m = \frac{1 \pm 3}{4} \]
This gives us two values:
\[ m_1 = \frac{4}{4} = 1 \quad \text{and} \quad m_2 = \frac{-2}{4} = -\frac{1}{2} \]

### Step 5: Calculate \( \tan \alpha \)
Using the formula for the tangent of the angle between two lines with slopes \( m_1 \) and \( m_2 \):
\[ \tan \alpha = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \( m_1 = 1 \) and \( m_2 = -\frac{1}{2} \):
\[ \tan \alpha = \left| \frac{1 - (-\frac{1}{2})}{1 + (1)(-\frac{1}{2})} \right| = \left| \frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} \right| = \left| \frac{\frac{3}{2}}{\frac{1}{2}} \right| = 3 \]

### Final Answer
Thus, the value of \( \tan \alpha \) is:
\[ \boxed{3} \]