Let `A(x_(1),y_(1))` and `B(x_(2),y_(2))` be two points on the parabola `y^(2) = 4ax`. If the circle with chord AB as a dimater touches the parabola, then `|y_(1)-y_(2)|` is equal to

Options

A. `4a`
B. `8a`
C. `6sqrt(2)a`
D. not a constant

Explanation

To solve the problem, we need to find the absolute difference between the y-coordinates of two points on the parabola \( y^2 = 4ax \) given that the circle with chord AB as a diameter touches the parabola.

### Step-by-Step Solution:

1. **Identify Points on the Parabola**:
The points \( A \) and \( B \) on the parabola can be expressed in terms of parameters \( t_1 \) and \( t_2 \):
\[ A(t_1) = (at_1^2, 2at_1), \quad B(t_2) = (at_2^2, 2at_2) \]

2. **Equation of the Circle**:
The circle with diameter \( AB \) has its center at the midpoint of \( A \) and \( B \). The equation of the circle can be written as:
\[ \left(x - \frac{at_1^2 + at_2^2}{2}\right)^2 + \left(y - \frac{2at_1 + 2at_2}{2}\right)^2 = \left(\frac{d}{2}\right)^2 \]
where \( d \) is the distance between points \( A \) and \( B \).

3. **Finding the Condition for Tangency**:
For the circle to touch the parabola, we need to substitute the parametric equations of the parabola into the equation of the circle and ensure that the resulting equation has a double root (discriminant \( D = 0 \)).

4. **Substituting into the Circle Equation**:
Substitute \( x = at^2 \) and \( y = 2at \) into the circle's equation. This leads to a quadratic equation in \( t \):
\[ (t^2 - t_1^2)(t^2 - t_2^2) + 4a(t - t_1)(t - t_2) = 0 \]

5. **Condition for Double Roots**:
The discriminant of this quadratic must be zero for the circle to touch the parabola:
\[ (t_1 + t_2)^2 - 4(t_1 t_2 + 4) = 0 \]

6. **Solving the Discriminant**:
Expanding and simplifying gives:
\[ t_1^2 + t_2^2 - 2t_1t_2 - 16 = 0 \]
This can be rearranged to:
\[ (t_1 - t_2)^2 = 16 \implies |t_1 - t_2| = 4 \]

7. **Finding \( |y_1 - y_2| \)**:
The y-coordinates are given by:
\[ y_1 = 2at_1, \quad y_2 = 2at_2 \]
Therefore, the absolute difference is:
\[ |y_1 - y_2| = |2at_1 - 2at_2| = 2a|t_1 - t_2| = 2a \cdot 4 = 8a \]

8. **Conclusion**:
Thus, the final answer is:
\[ |y_1 - y_2| = 8a \]