From an external point P tangents are drawn to the parabola, find the equation to the locus of P when these tangents make angles ` theta _(1) and theta _(2)` with the axis, such that
` tan theta _(1) tan theta _(2) ` is constant (= c)

Answer

To find the equation of the locus of point P from which tangents are drawn to the parabola \( y^2 = 4ax \), we will follow these steps:

### Step 1: Write the equation of the tangents
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point where the slope is \( m \) is given by:
\[ y = mx + \frac{a}{m} \]

### Step 2: Set up the point P
Let point P be represented as \( P(h, k) \). The point P lies on both tangents drawn to the parabola. Therefore, it must satisfy the tangent equation for both slopes \( m_1 \) and \( m_2 \).

### Step 3: Substitute point P into the tangent equation
For the first tangent with slope \( m_1 \):
\[ k = m_1h + \frac{a}{m_1} \]
For the second tangent with slope \( m_2 \):
\[ k = m_2h + \frac{a}{m_2} \]

### Step 4: Rearranging the equations
Rearranging both equations gives us:
\[ m_1h - k + \frac{a}{m_1} = 0 \quad \text{(1)} \]
\[ m_2h - k + \frac{a}{m_2} = 0 \quad \text{(2)} \]

### Step 5: Form a quadratic equation
From equations (1) and (2), we can form a quadratic equation in terms of \( m \):
\[ m^2h - km + a = 0 \]
Here, \( m_1 \) and \( m_2 \) are the roots of this quadratic equation.

### Step 6: Use Vieta's formulas
From Vieta's formulas, we know:
- The sum of the roots \( m_1 + m_2 = \frac{k}{h} \)
- The product of the roots \( m_1 m_2 = \frac{a}{h} \)

### Step 7: Given condition
We are given that \( \tan \theta_1 \tan \theta_2 = c \), which implies:
\[ m_1 m_2 = c \]
Thus, we have:
\[ \frac{a}{h} = c \]

### Step 8: Solve for h
From the equation \( \frac{a}{h} = c \), we can rearrange to find \( h \):
\[ h = \frac{a}{c} \]

### Step 9: Write the locus equation
The locus of point P can be expressed as:
\[ x = \frac{a}{c} \]
This is the equation of a vertical line.

### Final Answer
The equation of the locus of point P is:
\[ x = \frac{a}{c} \]

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