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 ^(2) theta _(1) = tan ^(2) theta _(2)` is constant ` ( = lambda )`

Answer

To find the equation of the locus of point P from which tangents are drawn to the parabola \( y^2 = 4ax \), and given that the tangents make angles \( \theta_1 \) and \( \theta_2 \) with the x-axis such that \( \tan^2 \theta_1 = \tan^2 \theta_2 = \lambda \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parabola Equation**:
The equation of the parabola is given as:
\[ y^2 = 4ax \]

2. **Equation of Tangent in Slope Form**:
The equation of the tangent to the parabola in slope form is:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

3. **Coordinates of External Point P**:
Let the coordinates of point P be \( (h, k) \). For the tangents to pass through this point, we substitute \( (h, k) \) into the tangent equation:
\[ k = mh + \frac{a}{m} \]

4. **Rearranging to Form a Quadratic Equation**:
Rearranging the above equation gives:
\[ km = mh^2 + a \]
Multiplying through by \( m \) leads to:
\[ m^2h - km + a = 0 \]
This is a quadratic equation in \( m \).

5. **Roots of the Quadratic**:
Let \( m_1 \) and \( m_2 \) be the slopes of the tangents. By Vieta's formulas, we have:
\[ m_1 + m_2 = \frac{k}{h} \quad \text{and} \quad m_1 m_2 = \frac{a}{h} \]

6. **Using the Given Condition**:
Given that \( \tan^2 \theta_1 = \tan^2 \theta_2 = \lambda \), we have:
\[ m_1^2 = m_2^2 = \lambda \]
This implies \( m_1 = \sqrt{\lambda} \) and \( m_2 = -\sqrt{\lambda} \) (or vice versa).

7. **Sum of Squares of Slopes**:
From the identity:
\[ m_1^2 + m_2^2 = (m_1 + m_2)^2 - 2m_1 m_2 \]
We can substitute:
\[ 2\lambda = \left(\frac{k}{h}\right)^2 - 2\left(\frac{a}{h}\right) \]

8. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{k^2}{h^2} - \frac{2a}{h} - 2\lambda = 0 \]
Multiplying through by \( h^2 \) leads to:
\[ k^2 - 2ah - 2\lambda h^2 = 0 \]

9. **Locus Equation**:
This is the equation of the locus of point P, which can be rewritten as:
\[ k^2 = 2\lambda h^2 + 2ah \]

### Final Equation:
Thus, the equation of the locus of point P is:
\[ y^2 = 2\lambda x^2 + 2ax \]