If `theta` is the angle between the tangents to the parabola `y^(2)=12x` passing through the point (-1,2) then `|tan theta|` is equal to

Options

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

Answer

To find the value of \(|\tan \theta|\), where \(\theta\) is the angle between the tangents to the parabola \(y^2 = 12x\) passing through the point \((-1, 2)\), we will follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \(y^2 = 12x\). We can rewrite this in the standard form \(y^2 = 4ax\) to find \(a\):
\[ 4a = 12 \implies a = 3 \]

### Step 2: Write the equation of the tangent line
The equation of the tangent to the parabola \(y^2 = 4ax\) at a point with slope \(m\) is given by:
\[ y = mx + \frac{a}{m} \]
Substituting \(a = 3\):
\[ y = mx + \frac{3}{m} \]

### Step 3: Substitute the point into the tangent equation
We need the tangent to pass through the point \((-1, 2)\). Thus, substituting \(x = -1\) and \(y = 2\) into the tangent equation:
\[ 2 = m(-1) + \frac{3}{m} \]
This simplifies to:
\[ 2 = -m + \frac{3}{m} \]
Multiplying through by \(m\) to eliminate the fraction:
\[ 2m = -m^2 + 3 \]
Rearranging gives:
\[ m^2 + 2m - 3 = 0 \]

### Step 4: Solve the quadratic equation
We can solve the quadratic equation \(m^2 + 2m - 3 = 0\) using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 2\), and \(c = -3\):
\[ m = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \]
\[ m = \frac{-2 \pm \sqrt{4 + 12}}{2} \]
\[ m = \frac{-2 \pm \sqrt{16}}{2} \]
\[ m = \frac{-2 \pm 4}{2} \]
This gives us two solutions:
\[ m_1 = 1 \quad \text{and} \quad m_2 = -3 \]

### Step 5: Find the angle between the tangents
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting \(m_1 = 1\) and \(m_2 = -3\):
\[ \tan \theta = \frac{1 - (-3)}{1 + (1)(-3)} = \frac{1 + 3}{1 - 3} = \frac{4}{-2} = -2 \]

### Step 6: Find \(|\tan \theta|\)
Thus, we find:
\[ |\tan \theta| = 2 \]

### Final Answer
\[ |\tan \theta| = 2 \]

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