If the normals to the parabola `y^2=4a x` at three points `(a p^2,2a p),` and `(a q^2,2a q)` are concurrent, then the common root of equations `P x^2+q x+r=0` and `a(b-c)x^2+b(c-a)x+c(a-b)=0` is `p` (b) `q` (c) `r` (d) `1`

Options

A. p
B. q
C. r
D. 1

Explanation

To solve the problem, we need to analyze the conditions given for the normals to the parabola \( y^2 = 4ax \) at the points \( (ap^2, 2ap) \), \( (aq^2, 2aq) \), and \( (ar^2, 2ar) \). The normals at these points are said to be concurrent, which leads us to derive a relationship among \( p \), \( q \), and \( r \).

### Step-by-Step Solution:

1. **Identify the points on the parabola**:
The points given on the parabola are:
- \( P_1 = (ap^2, 2ap) \)
- \( P_2 = (aq^2, 2aq) \)
- \( P_3 = (ar^2, 2ar) \)

2. **Equation of the normal**:
The equation of the normal to the parabola \( y^2 = 4ax \) at the point \( (x_0, y_0) \) is given by:
\[ y - y_0 = -\frac{y_0}{2a}(x - x_0) \]
For point \( P_1 \):
\[ y - 2ap = -\frac{2ap}{2a}(x - ap^2) \Rightarrow y - 2ap = -p(x - ap^2) \]
Rearranging gives us:
\[ y = -px + ap^3 + 2ap \]

Similarly, we can derive the equations for the normals at \( P_2 \) and \( P_3 \).

3. **Concurrent normals condition**:
The normals are concurrent if the sum of the coefficients of \( x \) from the three normal equations is zero. This leads to:
\[ p + q + r = 0 \]

4. **Finding the common root**:
We need to find the common root of the equations:
\[ Px^2 + Qx + R = 0 \]
and
\[ a(b-c)x^2 + b(c-a)x + c(a-b) = 0 \]
Since we know \( p + q + r = 0 \), we can substitute \( r = -p - q \).

5. **Testing for the root**:
We can check if \( x = 1 \) is a root of the first equation:
\[ P(1)^2 + Q(1) + R = P + Q + R \]
Since \( P + Q + R = 0 \), \( x = 1 \) is indeed a root.

6. **Check the second equation**:
Now we check if \( x = 1 \) is also a root of the second equation:
\[ a(b-c)(1)^2 + b(c-a)(1) + c(a-b) = a(b-c) + b(c-a) + c(a-b) \]
Expanding this gives:
\[ ab - ac + bc - ba + ca - cb = 0 \]
This simplifies to \( 0 = 0 \), confirming \( x = 1 \) is a root.

### Conclusion:
Thus, the common root of both equations is \( 1 \).