Question

If the normal at \( (a^2, 2a^2) \) on the parabola \( y^2 = 4ax \), meets the parabola again at \( (a^2, 2a^2) \), then

• A. \( p^2 + pq + 2 = 0 \)
• B. \( p^2 - pq + 2 = 0 \)
• C. \( q^2 + pq + 2 = 0 \)
• D. \( p^2 + pq + 1 = 0 \)

Hint:

For parabolas, the equation of the normal at any point can be used to derive relationships between coordinates and slopes.

Answer 1

The Correct Option is A

Step 1: Equation of normal to the parabola.
The equation of the normal to the parabola \( y^2 = 4ax \) at any point \( (x_1, y_1) \) is derived, and solving for \( p \) and \( q \) yields the equation \( p^2 + pq + 2 = 0 \).

Step 2: Conclusion.
Thus, the correct answer is option (A).

Final Answer: \[ \boxed{\text{(A) } p^2 + pq + 2 = 0} \]