---
title: "Tangents from centre of hyperbola are asymptotes"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:hyperbola-tangents-from-centre-asymptotes"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/hyperbola-tangents-from-centre-asymptotes
site: BCT Engineering
---

# Tangents from centre of hyperbola are asymptotes

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2072 Kartik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/hyperbola-tangents-from-centre-asymptotes
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Show that the pair of tangents drawn from the center of a hyperbola are its asymptotes.

## Short answer

Tangents from the centre (0,0) to x²/a² − y²/b² = 1 satisfy c=0 in c² = a²m² − b², giving m = ±b/a. These lines y = ±(b/a)x are the asymptotes.

## Answer

An asymptote of a hyperbola is a tangent whose point of contact is at infinity and which passes through the centre. We show that the two tangents drawn from the centre are exactly the lines y = ±(b/a)x.

#### Setup
Take the hyperbola x²/a² − y²/b² = 1 with the centre at the origin O(0, 0). Consider a line through O: y = mx.

#### Tangent condition
A line y = mx + c touches the hyperbola if c² = a²m² − b². (Derivation: put y = mx + c in b²x² − a²y² = a²b² to get (b² − a²m²)x² − 2a²mc x − a²(c² + b²) = 0; equal roots give a²m²c² + (b² − a²m²)(c² + b²) = 0, i.e. c² = a²m² − b².)

For the line to pass through the centre, c = 0:

0 = a²m² − b² → m² = b²/a² → m = ± b/a

#### Conclusion
The tangents from the centre are y = (b/a)x and y = −(b/a)x, that is

x²/a² − y²/b² = 0

These are precisely the asymptotes of the hyperbola. (The point of contact is at infinity: with c = 0 and m² = b²/a², both the x² and x coefficients of the quadratic vanish, leaving −a²b² = 0, so both roots are infinite.) Hence the pair of tangents from the centre are its asymptotes.

## Check yourself

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

1. An asymptote is?
   A. A tangent through the centre whose point of contact is at infinity
   B. A line through a focus
   C. A chord through the vertex
   D. The directrix
   Answer: A. That is the definition used. (Key point: Asymptote idea)

2. For x^2/a^2 − y^2/b^2 = 1, y = mx + c touches if?
   A. c^2 = a^2m^2 + b^2
   B. c = am − b
   C. c^2 = b^2m^2 − a^2
   D. c^2 = a^2m^2 − b^2
   Answer: D. Equal roots condition. (Key point: Tangent condition)

3. Putting c = 0 gives?
   A. m = ± a/b
   B. m = 0
   C. m = ± b/a
   D. m = ± ab
   Answer: C. 0 = a^2m^2 − b^2. (Key point: Passing through centre)

4. Why is the contact at infinity?
   A. The roots are zero
   B. With c = 0 the leading coefficient b^2 − a^2m^2 is zero, so both roots are infinite
   C. The line is vertical
   D. The discriminant is negative
   Answer: B. The quadratic degenerates. (Key point: Contact at infinity)

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