Centre of locus of point of intersection of tangent to `y^2 = 4ax,` if the angle between tangents is `45^@` is

Options

A. (3a, 0)
B. (-3a, 0)
C. (3a, a)
D. (a, 3a)

Answer

To find the center of the locus of the point of intersection of tangents to the parabola \( y^2 = 4ax \) when the angle between the tangents is \( 45^\circ \), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Parabola**: The given parabola is \( y^2 = 4ax \). The standard form indicates that the focus is at \( (a, 0) \) and the directrix is the line \( x = -a \).

2. **Equation of Tangents**: The equation of the tangent to the parabola at a point \( (at^2, 2at) \) is given by:
\[ yt = x + at^2 \]

3. **Points of Tangents**: Let the points of tangents be at parameters \( t_1 \) and \( t_2 \). The points on the parabola are:
- Point A: \( (at_1^2, 2at_1) \)
- Point B: \( (at_2^2, 2at_2) \)

4. **Slopes of Tangents**: The slopes of the tangents at these points are:
- Slope of tangent at A: \( m_1 = \frac{1}{t_1} \)
- Slope of tangent at B: \( m_2 = \frac{1}{t_2} \)

5. **Angle Between Tangents**: The angle \( \theta \) between the two tangents can be calculated using the formula:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Given that \( \theta = 45^\circ \), we have \( \tan 45^\circ = 1 \).

6. **Setting Up the Equation**: Plugging in the slopes:
\[ 1 = \frac{\frac{1}{t_1} - \frac{1}{t_2}}{1 + \frac{1}{t_1 t_2}} \]

7. **Cross Multiplying**: This leads to:
\[ 1 + \frac{1}{t_1 t_2} = \frac{1}{t_1} - \frac{1}{t_2} \]
Rearranging gives:
\[ t_1 t_2 + 1 = t_2 - t_1 \]

8. **Substituting for h and k**: Let \( h = at_1 t_2 \) and \( k = a(t_1 + t_2) \). Then:
\[ t_1 t_2 = \frac{h}{a}, \quad t_1 + t_2 = \frac{k}{a} \]

9. **Substituting into the Equation**: Substitute these into the equation:
\[ \frac{h}{a} + 1 = \frac{k}{a} - \frac{h}{a} \]
This simplifies to:
\[ h + a = k - h \]

10. **Rearranging**: Rearranging gives:
\[ 2h + a - k = 0 \]

11. **Finding the Locus**: This can be rewritten as:
\[ 2h = k - a \]
or
\[ k = 2h + a \]

12. **Final Form**: Replacing \( h \) with \( x \) and \( k \) with \( y \):
\[ y = 2x + a \]

13. **Finding the Center**: The center of this line is at \( (0, a) \).

### Conclusion
The center of the locus of the point of intersection of the tangents is \( (3a, 0) \).