Question

If the normal to the parabola $y²=4ax$ at the point $(at², 2at)$ cuts the parabola again at $(aT², 2aT)$, then

• A. $-2 \le T \le 2$
• B. $T \in (-\infty, -8) \cup (8, \infty)$
• C. $T^2<8$
• D. $T^2 \ge 8$

Hint:

If the normal to the parabola $y=4ax$ at the point $(at, 2at)$ cuts the parabola again at $(aT, 2aT)$, then

Answer 1

The Correct Option is D

Step 1: Concept
The relation between parameters $t$ and $T$ for a normal chord is $T = -t - 2/t$.
Step 2: Analysis
Rearranging gives the quadratic $t^2 + tT + 2 = 0$.
Step 3: Evaluation
For $t$ to be a real parameter, the discriminant $D = T^2 - 4(1)(2)$ must be non-negative.
Step 4: Conclusion
$T^2 - 8 \ge 0 \Rightarrow T^2 \ge 8$.
Final Answer: (d)