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Coordinate Geometry · Mathematics

Common Tangents to Two Circles Simulation

Drag two circles apart, together and inside each other, then draw every common tangent. Watch the count go 4 → 3 → 2 → 1 → 0 as the distance between the centres d crosses r₁ + r₂ and |r₁ − r₂|.

Interactive virtual lab · no download · works on phones and laptops

Chapter
Circles
For
Class 11 · JEE
Time
~10 min

Common Tangents to Two Circles is a Pro simulation

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How to use it

  1. 1Drag a centre to move a circle; drag the ring handle to resize it.
  2. 2Predict the number, then tick “Draw common tangents to both the circles”.
  3. 3Use the arrangement buttons to jump to each case.
  4. 4Take the Quiz; after each answer, “Show me” sets the case up for you.

What to notice

  • d > r₁ + r₂: four tangents (two direct, two transverse).
  • d = r₁ + r₂: three. The transverse pair merges at the point of contact.
  • |r₁ − r₂| < d < r₁ + r₂: two, both direct.
  • d = |r₁ − r₂|: one, at the point where they touch internally.
  • d < |r₁ − r₂|: none. One circle hides inside the other.
  • Transverse tangents meet where C₁C₂ is divided internally in r₁ : r₂; direct ones externally.

For teachers

A ready-made teaching kit comes with this lab

  • Lesson plan

    20 min, 4 steps

  • Ask the class

    3 questions with answers

  • Quick quiz

    7 questions, printable

  • Key formulas

    3 formulas

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About the Common Tangents to Two Circles simulation

Two circles can have up to four common tangents, and how many depends on how far apart their centres are compared with their radii: four if they are separate, three if they touch externally, two if they cut, one if they touch internally and none if one lies inside the other. The direct tangents meet at the external centre of similitude and the transverse ones at the internal centre.

In this common tangents simulation you drag two circles, change their radii, snap them to touching, and see every common tangent drawn with the lengths of the direct and transverse tangents and both centres of similitude. It covers Circles in Class 11 coordinate geometry and JEE.

Also covers: direct common tangent, transverse common tangent, number of common tangents, centre of similitude, two circles.

Frequently asked questions

How many common tangents do two circles touching externally have?

Three: two direct tangents and one transverse tangent at the point of contact.

When do two circles have no common tangent?

When one lies entirely inside the other: d < |r₁ − r₂|.

What condition gives four common tangents?

The circles are separate: d > r₁ + r₂.

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