---
title: "Normal to rectangular hyperbola xy=c²"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rectangular-hyperbola-normal-t-t1"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rectangular-hyperbola-normal-t-t1
site: BCT Engineering
---

# Normal to rectangular hyperbola xy=c²

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2068 Chaitra
- Video: [Lecture 19 Rectangular hyperbola xy=c2: Characteristics & proof: Chord, tangent & normal equation](https://www.youtube.com/watch?v=O4-NT7dhUEw) — Mathsmerizing
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/rectangular-hyperbola-normal-t-t1
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Prove that the normal at a point t of the rectangular hyperbola xy = c² meets the curve again at a point t₁ such that t³t₁ = −1.

## Short answer

The normal at t (ct, c/t) on xy = c² has slope t². Substituting the second intersection t₁ gives t³ t₁ = −1.

## Answer

For the rectangular hyperbola xy = c², a point t is (ct, c/t).

#### Slope of tangent and normal
Differentiating xy = c²: x dy/dx + y = 0, so dy/dx = −y/x = −(c/t)/(ct) = −1/t².

The normal has slope t² (negative reciprocal).

#### Equation of the normal at t
y − c/t = t²(x − ct)

#### Second intersection
Let the normal meet the curve again at t₁, the point (ct₁, c/t₁). Substitute:

c/t₁ − c/t = t²(ct₁ − ct)

c(t − t₁)/(t t₁) = c t²(t₁ − t)

Since t ≠ t₁, divide by c(t − t₁):

1/(t t₁) = −t²

**t³ t₁ = −1**

Hence the normal at t meets the curve again at the point t₁ = −1/t³. Proved.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which point represents the parameter t on xy = c^2?
   A. (c/t, ct^2)
   B. (c/t, c/t)
   C. (ct^2, c/t^2)
   D. (ct, c/t)
   Answer: D. Substituting x = ct and y = c/t gives xy = c^2, so (ct, c/t) lies on the curve. (Key point: Parametric point)

2. The tangent slope at t is -1/t^2. What is the slope of the normal?
   A. -1/t^2
   B. -t^2
   C. t^2
   D. 1/t^2
   Answer: C. The normal slope is the negative reciprocal of the tangent slope, giving t^2. (Key point: Slope of the normal)

3. After substituting (ct1, c/t1) into the normal, what relation results once c(t - t1) is cancelled (t != t1)?
   A. 1/(t t1) = t^2
   B. 1/(t t1) = -t^2
   C. t t1 = -t^2
   D. t1 - t = -t^2
   Answer: B. c(t - t1)/(t t1) = c t^2 (t1 - t) = -c t^2 (t - t1), so 1/(t t1) = -t^2. (Key point: Condition at second point)

4. What does t^3 t1 = -1 give for the second meeting point?
   A. t1 = 1/t^3
   B. t1 = -1/t^3
   C. t1 = -t^3
   D. t1 = -1/t^2
   Answer: B. Dividing t^3 t1 = -1 by t^3 gives t1 = -1/t^3. (Key point: Final result)

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