Let P (a, b) and Q(c, d) are the two points on the parabola `y^2=8x` such that the normals at them meet in (18, 12). Then the product (abcd) is:

Options

A. 412
B. 410
C. 512
D. 510

Answer

To solve the problem, we need to find the product \( abcd \) given that points \( P(a, b) \) and \( Q(c, d) \) lie on the parabola \( y^2 = 8x \) and that the normals at these points meet at \( (18, 12) \).

### Step 1: Identify the parameters of the parabola
The equation of the parabola is given by \( y^2 = 8x \). We can rewrite this in the standard form \( y^2 = 4ax \) to identify \( a \):
\[ 4a = 8 \implies a = 2 \]

### Step 2: General point on the parabola
The general point on the parabola can be expressed as:
\[ P(t) = (2t^2, 4t) \]
where \( t \) is a parameter.

### Step 3: Equation of the normal
The equation of the normal at the point \( (2t^2, 4t) \) on the parabola is given by:
\[ y = -tx + 4t + 2t^3 \]

### Step 4: Substitute the point where normals meet
We know that the normals at points \( P \) and \( Q \) meet at \( (18, 12) \). We can substitute \( x = 18 \) and \( y = 12 \) into the normal equation:
\[ 12 = -t(18) + 4t + 2t^3 \]
This simplifies to:
\[ 12 = -18t + 4t + 2t^3 \implies 12 = -14t + 2t^3 \]
Rearranging gives:
\[ 2t^3 - 14t - 12 = 0 \implies t^3 - 7t - 6 = 0 \]

### Step 5: Finding the roots of the cubic equation
We can use the Rational Root Theorem to find possible rational roots. Testing \( t = 3 \):
\[ 3^3 - 7(3) - 6 = 27 - 21 - 6 = 0 \]
Thus, \( t = 3 \) is a root. We can factor the cubic polynomial:
\[ t^3 - 7t - 6 = (t - 3)(t^2 + 3t + 2) \]
Factoring further gives:
\[ t^2 + 3t + 2 = (t + 1)(t + 2) \]
Thus, the roots are:
\[ t = 3, t = -1, t = -2 \]

### Step 6: Finding points P and Q
Now we can find the points corresponding to these values of \( t \):
1. For \( t = 3 \):
\[ P(3) = (2(3^2), 4(3)) = (18, 12) \]
(This point is not valid since it is where the normals meet.)

2. For \( t = -1 \):
\[ P(-1) = (2(-1)^2, 4(-1)) = (2, -4) \]

3. For \( t = -2 \):
\[ P(-2) = (2(-2)^2, 4(-2)) = (8, -8) \]

### Step 7: Assigning values to \( a, b, c, d \)
From the points we have:
- \( P = (8, -8) \) gives \( a = 8, b = -8 \)
- \( Q = (2, -4) \) gives \( c = 2, d = -4 \)

### Step 8: Calculating the product \( abcd \)
Now we can calculate the product:
\[ abcd = (8)(-8)(2)(-4) \]
Calculating this step-by-step:
\[ = 64 \cdot 8 = 512 \]

### Final Answer
Thus, the product \( abcd \) is:
\[ \boxed{512} \]