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 ( = b)

Answer

To find the equation of the locus of point P from which tangents are drawn to the parabola, we will follow these steps:

### Step-by-Step Solution:

1. **Assume the Coordinates of Point P**:
Let the external point P have coordinates (h, k).

**Hint**: Start by defining the point from which the tangents are drawn.

2. **Equation of Tangents to the Parabola**:
The equation of the tangents to the parabola \(y^2 = 4ax\) from point P can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \(m\) is the slope of the tangent.

**Hint**: Recall the general form of the tangent to a parabola.

3. **Substituting Point P into the Tangent Equation**:
Since point P lies on the tangents, we substitute (h, k) into the tangent equation:
\[ k = mh + \frac{a}{m} \]
Rearranging gives:
\[ km = m^2h + a \]

**Hint**: Use the relationship between the point and the tangent to derive a quadratic equation.

4. **Forming a Quadratic Equation**:
Rearranging the equation gives:
\[ m^2h - km + a = 0 \]
This is a quadratic equation in \(m\).

**Hint**: Recognize that this quadratic will help us find relationships between the slopes of the tangents.

5. **Using Vieta's Formulas**:
From Vieta's formulas, for the roots \(m_1\) and \(m_2\) of the quadratic:
- The sum of the slopes \(m_1 + m_2 = \frac{k}{h}\)
- The product of the slopes \(m_1 m_2 = \frac{a}{h}\)

**Hint**: Relate the coefficients of the quadratic to the slopes of the tangents.

6. **Given Condition**:
We are given that \( \tan \theta_1 + \tan \theta_2 = b \), which implies:
\[ m_1 + m_2 = b \]
Therefore, we have:
\[ \frac{k}{h} = b \implies k = bh \]

**Hint**: Use the condition provided in the problem to relate \(k\) and \(h\).

7. **Equation of the Locus**:
The equation \(k = bh\) represents the locus of point P. This can be rewritten as:
\[ y = bx \]

**Hint**: Recognize that this is the equation of a straight line, which represents the locus.

### Final Answer:
The equation of the locus of point P is:
\[ y = bx \]