Find the locus of a point O when the three normals drawn from it are such that
two of them make equal angles with the given line `y = mx = c`

Answer

To find the locus of a point \( O \) from which three normals are drawn to a parabola such that two of them make equal angles with the given line \( y = mx + c \), we can follow these steps:

### Step 1: Understand the Geometry
The problem states that two normals from point \( O \) make equal angles with the line \( y = mx + c \). This means that the slopes of these two normals will be equal.

### Step 2: Set Up the Slopes
Let the slopes of the normals from point \( O \) be \( M_1 \), \( M_2 \), and \( M_3 \). Since two normals make equal angles with the line, we can assume \( M_1 = M_2 = M \).

### Step 3: Use the Angle Between Two Lines Formula
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Since \( M_1 = M_2 \), the angle between them becomes zero, which means they are parallel.

### Step 4: Write the Equation of the Normal
The equation of the normal to the parabola \( y^2 = 4ax \) at a point with slope \( M \) can be expressed as:
\[ y = Mx - 2aM - aM^3 \]
Rearranging gives:
\[ aM^3 + 2aM - xM + y = 0 \]

### Step 5: Substitute the Coordinates of Point \( O \)
Let the coordinates of point \( O \) be \( (h, k) \). The point \( O \) must satisfy the cubic equation:
\[ aM^3 + 2aM - hM + k = 0 \]

### Step 6: Analyze the Roots of the Cubic Equation
The roots of the cubic equation are \( M_1, M_2, M_3 \). Since \( M_1 = M_2 = M \), we can express the product of the roots:
\[ M_1 M_2 M_3 = -\frac{k}{a} \]
This gives us:
\[ M^2 M_3 = -\frac{k}{a} \quad \text{(1)} \]

### Step 7: Sum of the Roots
The sum of the roots is given by:
\[ M_1 + M_2 + M_3 = 0 \]
This implies:
\[ 2M + M_3 = 0 \quad \Rightarrow \quad M_3 = -2M \quad \text{(2)} \]

### Step 8: Substitute into the Product of Roots
Substituting (2) into (1):
\[ M^2 (-2M) = -\frac{k}{a} \]
This simplifies to:
\[ -2M^3 = -\frac{k}{a} \quad \Rightarrow \quad k = 2aM^3 \quad \text{(3)} \]

### Step 9: Use the Relationship Between \( M \) and \( h \)
From the relationship between the roots, we also have:
\[ M_1 + M_2 = 2M = \frac{h}{a} - 2M \]
This leads to:
\[ 4M = \frac{h}{a} \quad \Rightarrow \quad M = \frac{h}{4a} \quad \text{(4)} \]

### Step 10: Substitute \( M \) into \( k \)
Substituting (4) into (3):
\[ k = 2a\left(\frac{h}{4a}\right)^3 = 2a\left(\frac{h^3}{64a^3}\right) = \frac{h^3}{32a^2} \]

### Final Step: Locus Equation
The locus of point \( O \) can be expressed as:
\[ k = \frac{h^3}{32a^2} \]
This is the equation of a parabola.

### Conclusion
The locus of the point \( O \) is a parabola given by the equation \( k = \frac{h^3}{32a^2} \).

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