Angle between tangents drawn from the point `(1, 4)` to the parabola `y^2 = 4x` is

Options

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

Answer

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

### Step 1: Write the equation of the parabola
The given parabola is \( y^2 = 4x \).

### Step 2: Use the formula for the equation of the tangent
The equation of the tangent to the parabola \( y^2 = 4ax \) at the point where the slope of the tangent is \( m \) is given by:
\[ y = mx + \frac{a}{m} \]
Here, \( a = 1 \) (since \( 4a = 4 \Rightarrow a = 1 \)). Thus, the equation becomes:
\[ y = mx + \frac{1}{m} \]

### Step 3: Substitute the point (1, 4) into the tangent equation
We need to find the slopes \( m \) such that the tangent passes through the point (1, 4):
\[ 4 = m(1) + \frac{1}{m} \]
This simplifies to:
\[ 4 = m + \frac{1}{m} \]

### Step 4: Multiply through by \( m \) to eliminate the fraction
Multiplying both sides by \( m \) gives:
\[ 4m = m^2 + 1 \]
Rearranging this gives us a quadratic equation:
\[ m^2 - 4m + 1 = 0 \]

### Step 5: Solve the quadratic equation for \( m \)
Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -4, c = 1 \):
\[ m = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Thus, the slopes of the tangents are \( m_1 = 2 + \sqrt{3} \) and \( m_2 = 2 - \sqrt{3} \).

### Step 6: Find \( m_1 - m_2 \) and \( m_1 m_2 \)
Calculating \( m_1 - m_2 \):
\[ m_1 - m_2 = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2\sqrt{3} \]
Calculating \( m_1 m_2 \):
\[ m_1 m_2 = (2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \]

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

### Step 8: Find the angle \( \theta \)
Since \( \tan \theta = \sqrt{3} \), we know that:
\[ \theta = 60^\circ = \frac{\pi}{3} \text{ radians} \]

### Conclusion
The angle between the tangents drawn from the point (1, 4) to the parabola \( y^2 = 4x \) is \( \frac{\pi}{3} \).

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