Two tangents to parabola `y ^(2 ) = 4ax` make angles `theta and phi` with the x-axis. Then find the locus of their point of intersection if `sin (theta - phi) =2 cos theta cos phi.`

Answer

To find the locus of the point of intersection of two tangents to the parabola \( y^2 = 4ax \) that make angles \( \theta \) and \( \phi \) with the x-axis, given the condition \( \sin(\theta - \phi) = 2 \cos \theta \cos \phi \), we can follow these steps:

### Step 1: Understand the Tangents to the Parabola
The equation of the parabola is \( y^2 = 4ax \). The slope of the tangent to this parabola at any point can be expressed as \( m \) where \( m = \frac{y}{2a} \). The equation of the tangent line at the point \( (at^2, 2at) \) is given by:
\[ y = mx - 2am \]

### Step 2: Write the Tangents for Angles \( \theta \) and \( \phi \)
For the angles \( \theta \) and \( \phi \), the slopes of the tangents can be expressed as:
\[ m_1 = \tan \theta \quad \text{and} \quad m_2 = \tan \phi \]
Thus, the equations of the tangents can be written as:
\[ y = \tan \theta \cdot x - 2a \tan \theta \quad \text{(1)} \]
\[ y = \tan \phi \cdot x - 2a \tan \phi \quad \text{(2)} \]

### Step 3: Find the Point of Intersection
To find the point of intersection of these two tangents, we set the right-hand sides of equations (1) and (2) equal to each other:
\[ \tan \theta \cdot x - 2a \tan \theta = \tan \phi \cdot x - 2a \tan \phi \]
Rearranging gives:
\[ (\tan \theta - \tan \phi)x = 2a(\tan \theta - \tan \phi) \]
Assuming \( \tan \theta \neq \tan \phi \), we can divide both sides by \( \tan \theta - \tan \phi \):
\[ x = 2a \]

### Step 4: Substitute \( x \) into the Tangent Equations
Now, substituting \( x = 2a \) back into either tangent equation to find \( y \):
Using equation (1):
\[ y = \tan \theta \cdot (2a) - 2a \tan \theta = 0 \]
Thus, the point of intersection is \( (2a, 0) \).

### Step 5: Find the Locus
Since \( a \) is a variable that can change, we can express the locus of the point of intersection as:
\[ x = 2a \implies a = \frac{x}{2} \]
The locus is thus:
\[ y = 0 \]
This means the locus of the point of intersection of the tangents is the x-axis.

### Final Answer
The locus of the point of intersection of the two tangents to the parabola \( y^2 = 4ax \) is the x-axis, represented by the equation:
\[ y = 0 \]