If two normals to `y^2=4ax` are perpendicular to each other, then the chords joining their feet are concurrent at the point :

Options

A. (a,0)
B. (2a,0)
C. (3a,0)
D. (4a,0)

Answer

To solve the problem, we need to find the point of concurrency of the chords joining the feet of the normals to the parabola \( y^2 = 4ax \) at points \( P \) and \( Q \), given that the normals are perpendicular to each other.

### Step 1: Parametric Representation of Points on the Parabola
Let the points \( P \) and \( Q \) on the parabola be represented in parametric form:
- For point \( P \): \( P(t_1) = (at_1^2, 2at_1) \)
- For point \( Q \): \( Q(t_2) = (at_2^2, 2at_2) \)

### Step 2: Slopes of the Normals
The slope of the normal at point \( P \) is given by:
\[ m_P = -t_1 \]
The slope of the normal at point \( Q \) is given by:
\[ m_Q = -t_2 \]

### Step 3: Condition for Perpendicular Normals
Since the normals are perpendicular, the product of their slopes must equal \(-1\):
\[ m_P \cdot m_Q = (-t_1)(-t_2) = t_1 t_2 = -1 \]

### Step 4: Coordinates of the Feet of the Normals
The equations of the normals at points \( P \) and \( Q \) can be written as:
- Normal at \( P \): \( y - 2at_1 = -t_1(x - at_1^2) \)
- Normal at \( Q \): \( y - 2at_2 = -t_2(x - at_2^2) \)

### Step 5: Finding the Intersection of the Normals
To find the feet of the normals, we need to solve these two equations simultaneously. The intersection point \( R(h, k) \) will lie on both normals.

### Step 6: Slope of the Chord PQ
The slope of the chord \( PQ \) is given by:
\[ \text{slope of } PQ = \frac{2at_1 - 2at_2}{at_1^2 - at_2^2} = \frac{2a(t_1 - t_2)}{a(t_1^2 - t_2^2)} = \frac{2(t_1 - t_2)}{t_1 + t_2} \]

### Step 7: Slope of PR
The slope of the line \( PR \) is given by:
\[ \text{slope of } PR = \frac{2at_1 - k}{at_1^2 - h} \]

### Step 8: Setting the Slopes Equal
Since point \( R \) lies on line \( PQ \), we can set the slopes equal:
\[ \frac{2(t_1 - t_2)}{t_1 + t_2} = \frac{2at_1 - k}{at_1^2 - h} \]

### Step 9: Cross-Multiplying and Simplifying
Cross-multiplying and simplifying the equation will yield a relationship between \( h \) and \( k \):
\[ 2a(t_1^2 - h) = (2at_1 - k)(t_1 - t_2) \]

### Step 10: Finding the Concurrent Point
After simplifying, we find that the concurrent point \( R \) is given by:
\[ h = a, \quad k = 0 \]

### Conclusion
Thus, the point of concurrency of the chords joining the feet of the normals is \( (a, 0) \).