Find the locus of a point O when the three normals drawn from it are such that
two of them make angles with the axis the product of whose tangents is 2.

Answer

To find the locus of a point \( O \) when the three normals drawn from it are such that two of them make angles with the axis whose tangents' product is 2, we can follow these steps:

### Step 1: Understand the given conditions
Let the point \( O \) be represented as \( (h, k) \). The slopes of the normals drawn from point \( O \) to the parabola will be denoted as \( m_1, m_2, \) and \( m_3 \). According to the problem, we know that:
\[ \tan \theta_1 \cdot \tan \theta_2 = 2 \]
This implies:
\[ m_1 \cdot m_2 = 2 \]

### Step 2: Write the equation of the normal to the parabola
The equation of the normal to a parabola \( y^2 = 4ax \) at a point with slope \( m \) is given by:
\[ y = mx - 2a(m - m^3) \]
We can rearrange this into a standard form:
\[ am^3 + 2a - mx + y = 0 \]

### Step 3: Substitute the point \( O(h, k) \)
Since the point \( O(h, k) \) lies on the normal, we substitute \( h \) and \( k \) into the equation:
\[ am^3 + 2a - hm + k = 0 \]
This is a cubic equation in terms of \( m \):
\[ am^3 - hm + (2a + k) = 0 \]

### Step 4: Identify the roots of the cubic equation
Let \( m_1, m_2, m_3 \) be the roots of the cubic equation. By Vieta's formulas, we have:
- The sum of the roots:
\[ m_1 + m_2 + m_3 = 0 \]
- The product of the roots:
\[ m_1 m_2 m_3 = -\frac{2a + k}{a} \]

### Step 5: Relate the roots
From the condition \( m_1 m_2 = 2 \), we can express \( m_3 \):
\[ m_3 = -\frac{2a + k}{a \cdot m_1 m_2} = -\frac{2a + k}{2a} \]

### Step 6: Use the sum of the roots
Using the sum of the roots:
\[ m_1 + m_2 = -m_3 \]
Substituting \( m_3 \):
\[ m_1 + m_2 = \frac{2a + k}{2a} \]

### Step 7: Use the product of the roots
Now, we can also express \( m_1 m_2 + m_2 m_3 + m_3 m_1 \):
\[ m_1 m_2 + m_2 m_3 + m_3 m_1 = 2 + m_3(m_1 + m_2) \]
Substituting \( m_1 + m_2 \):
\[ 2 + m_3 \cdot \frac{2a + k}{2a} = 2a - h \]

### Step 8: Solve for \( k \)
After substituting \( m_3 \) and simplifying, we can derive:
\[ 8a^2 - k^2 = 8a^2 - 4ah \]
This leads to:
\[ k^2 = 4ah \]

### Step 9: Final equation
Thus, we arrive at the equation:
\[ k^2 = 4ah \]
This is the equation of a parabola.

### Conclusion
The locus of the point \( O \) is given by the parabola:
\[ y^2 = 4ax \]