---
title: "Condition for lx+my+n=0 to be tangent to ellipse"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:ellipse-tangent-condition-lx-my-n"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-tangent-condition-lx-my-n
site: BCT Engineering
---

# Condition for lx+my+n=0 to be tangent to ellipse

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2067 Asar
- Video: [Class 12 Maths | Lecture 153 | Chapter 8 | Condition of Tangency of Line to Conic Sections](https://www.youtube.com/watch?v=pRizcrL3Hlw) — Saad Latif
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-tangent-condition-lx-my-n
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Find the condition that the line lx + my + n = 0 may be a tangent to the ellipse x²/a² + y²/b² = 1.

## Short answer

The line lx + my + n = 0 touches the ellipse x²/a² + y²/b² = 1 if a²l² + b²m² = n².

## Answer

The line lx + my + n = 0 touches the ellipse x²/a² + y²/b² = 1 if it meets the curve in two coincident points.

#### Method 1: using y = mx + c
For the ellipse, the line y = Mx + c is a tangent iff c² = a²M² + b².

Write lx + my + n = 0 as y = −(l/m)x − n/m. Here slope M = −l/m and c = −n/m. The condition becomes

n²/m² = a² l²/m² + b²

**a²l² + b²m² = n²**

#### Method 2: direct substitution (coincident roots)
Take y = −(lx + n)/m and put in b²x² + a²y² = a²b²:

b²m²x² + a²(lx + n)² = a²b²m²

(b²m² + a²l²)x² + 2a²ln x + a²(n² − b²m²) = 0

For equal roots, the discriminant is zero:

4a⁴l²n² − 4(b²m² + a²l²)a²(n² − b²m²) = 0

Divide by 4a²:

a²l²n² − (b²m²n² − b⁴m⁴ + a²l²n² − a²b²l²m²) = 0
b²m²(b²m² + a²l² − n²) = 0

so **a²l² + b²m² = n²**.

#### Point of contact
The point of contact is (−a²l/n, −b²m/n).

## Check yourself

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

1. When does a line touch the ellipse?
   A. It meets the curve in two coincident points
   B. It passes through the centre
   C. It is parallel to an axis
   D. It passes through a focus
   Answer: A. Tangency means equal roots (zero discriminant). (Key point: Tangent idea)

2. For y = Mx + c the tangent condition is?
   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. For the ellipse it is a^2M^2 + b^2. (Key point: Slope condition)

3. Condition for lx + my + n = 0 to touch?
   A. a^2/l^2 + b^2/m^2 = n^2
   B. a^2l^2 − b^2m^2 = n^2
   C. a^2l^2 + b^2m^2 = n^2
   D. l^2/a^2 + m^2/b^2 = n^2
   Answer: C. M = −l/m, c = −n/m gives n^2 = a^2l^2 + b^2m^2. (Key point: Result)

4. Point of contact?
   A. (a^2 l/n, b^2 m/n)
   B. (−a^2 l/n, −b^2 m/n)
   C. (−l/n, −m/n)
   D. (−a l/n, −b m/n)
   Answer: B. Obtained from the repeated root. (Key point: Contact point)

## Related in Mathematics I

- [Condition for lx+my+n=0 to be normal to ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-normal-condition-lx-my-n) — Mathematics I, chapter 3, asked 2×
- [Condition for x cosα + y sinα = p to touch ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-tangent-condition-xcos-ysin) — Mathematics I, chapter 3