---
title: "If the normal at point $t_1$ on the parabola $y^2 = 4ax$ meets the parabola again at $t_2$, then the relation between $t_1$ and $t_2$ is:"
canonical_url: "https://testx.in/questions/01a01eb9-4f2f-77c7-a566-449129d5ae82/if-the-normal-at-point-t1"
subject: "Mathematics"
topic: "Conic Sections"
difficulty: "HARD"
verified: false
cite_as: "TestX — Mathematics (Conic Sections): https://testx.in/questions/01a01eb9-4f2f-77c7-a566-449129d5ae82/if-the-normal-at-point-t1"
---
# If the normal at point $t_1$ on the parabola $y^2 = 4ax$ meets the parabola again at $t_2$, then the relation between $t_1$ and $t_2$ is:
- **Subject**: Mathematics
- **Topic**: Conic Sections
- **Difficulty**: HARD
- **Canonical URL**: https://testx.in/questions/01a01eb9-4f2f-77c7-a566-449129d5ae82/if-the-normal-at-point-t1
## Options
- **(A)** $t_1 t_2 = -1$
- **(B)** $t_1 + t_2 = 0$
- **(C)** $t_1^2 + t_1 t_2 + 2 = 0$
- **(D)** $t_1 t_2 = 4$
## Verified Answer
**Correct Answer**: Option (C): $t_1^2 + t_1 t_2 + 2 = 0$
### ⚡ 10-Second Exam Shortcut Trick
⚡ 10-Second Exam Shortcut: Eliminate options ending in incorrect units or calculate unit digits first before solving full arithmetic.
### 📝 Step-by-Step Derivation
1. Recall the property of normals meeting on the parabola $y^2 = 4ax$. If the normal at $t_1$ meets the parabola at $t_3$, then $t_3 = -t_1 - \frac{2}{t_1}$.
2. If the normal at $t_2$ also meets the parabola at the same point $t_3$, equate the expressions for $t_3$ corresponding to $t_1$ and $t_2$.
3. This leads to the fundamental relation $t_1 t_2 = 2$ (or $t_2 = -t_1 - 2/t_1$, simplifying to $t_1^2 + t_1 t_2 + t_2^2$ relationships depending on formulation; specifically $t_1 t_2 = 2$ or $t_1^2 + t_1 t_2 + t_2^2 + 2 = 0$). Wait, standard normal meeting condition on the curve is $t_2 = -t_1 - \frac{2}{t_1}$, giving $t_1^2 + t_1 t_2 + 2 = 0$.
### ⚠️ Common Trap to Avoid
⚠️ Common Trap: Misreading inverse proportions or rushing calculation in the last algebraic step.

### 💡 Exam Tip
💡 Exam Tip: Always verify if question asks for perimeter, area, or remaining fraction.

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