Coordinate Geometry · Mathematics

Circle Touching Two Circles: Locus 3D Simulation

The simulation of our “Two circles. One hidden curve.” video, in 3D on graph paper: a purple circle keeps squeezing between a big circle and a small one inside it while its centre leaves a green trail — pause at 85 % and predict, then watch the meter: PO₁ + PO₂ stays at R + r, so the curve is an ellipse. Pull the circles apart and the centre draws one branch of a hyperbola (PO₁ − PO₂ = R₁ − R₂), with the other branch belonging to circles that swallow both. Shrink the big circle and the branch straightens into the perpendicular bisector; then try every way a circle can touch two.

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

Chapter: Conic Sections
For: Class 11–12 · JEE Main · JEE Advanced
Time: ~20 min

Two Circles, One Hidden Curve 3D Lab is a Pro simulation

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How to use it

1Pick a module (4); the moving circle rolls by itself and draws its trail — answer the Predict to see the last 15 % and the meter.
2Drag the small circle's centre or use the sliders and buttons to change the circles.
3Use Derive, Questions, Practise and Compare; tap a live value marked ƒ to see its formula.

What to notice

Touching from outside: $P O = R + ρ$ ; from inside: $P O = R - ρ$ ; swallowing: $P O = ρ - R$ .
One inside the other, touched inside and outside: $P O_{1} + P O_{2} = R + r$ — an ellipse with foci at the centres.
Apart, touched from outside: $P O_{1} - P O_{2} = R_{1} - R_{2}$ — one branch of a hyperbola, nearer the small circle.
Equal radii: $P O_{1} = P O_{2}$ , the perpendicular bisector of $O_{1} O_{2}$ .
Opposite signs of $ρ$ cancel in the sum (ellipse); the same sign cancels in the difference (hyperbola).

For teachers

A ready-made teaching kit comes with this lab

Lesson plan
24 min, 4 steps
Ask the class
3 questions with answers
Quick quiz
5 questions, printable
Key formulas
4 formulas

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About the Two Circles, One Hidden Curve 3D Lab simulation

The Two Circles, One Hidden Curve 3D Lab is the simulation of our video of the same name. On graph paper in 3D, a big blue circle (radius 5) holds a small orange one (radius 1) inside it, and a purple circle keeps squeezing between them — inside the big one, outside the small one — while its centre P leaves a thick green trail. The trail stops at 85 % and asks you to predict the shape: a circle, an egg, or something else? Then the meter answers: PO₁ = R − ρ and PO₂ = r + ρ, so PO₁ + PO₂ = R + r = 6 whatever the moving circle's radius ρ, and the curve is an ellipse with the two centres as its foci (a = 3, b ≈ 2.73).

Pull the circles apart (radii 4 and 0.8, centres 6 apart) and touch both from outside: now subtracting cancels ρ, PO₁ − PO₂ = R₁ − R₂ = 3.2, and the centre draws one branch of a hyperbola — only one, because P is always nearer the small circle. The other branch appears when the moving circle swallows both. Shrink the big circle and the branch flattens into the perpendicular bisector when the radii are equal. The last module lets you choose the layout and the way each circle is touched, and explains when the result is an ellipse, a hyperbola branch, a line or impossible.

Also covers: locus of centre of circle touching two circles, circle touching two circles locus, locus of centre of a circle which touches two given circles, circles generate conic sections, ellipse as locus, hyperbola as locus, sum of focal distances, difference of focal distances, pf1 + pf2 = 2a, |pf1 - pf2| = 2a, one branch of hyperbola, perpendicular bisector locus, conic sections jee, locus jee, circles touching internally and externally.

Frequently asked questions

What is the locus of the centre of a circle that touches two given circles?

It depends on how it touches them. If one fixed circle lies inside the other and the moving circle touches the big one internally and the small one externally, the sum of the distances from its centre to the two fixed centres is constant (R + r), so the locus is an ellipse. If the fixed circles are apart and the moving circle touches both externally, the difference of distances is constant (R₁ − R₂), so the locus is one branch of a hyperbola.

Why only one branch of the hyperbola?

Because the difference has a fixed sign: PO₁ − PO₂ = R₁ − R₂ is positive, so P is always nearer the smaller circle's centre. The other branch, where PO₂ − PO₁ = R₁ − R₂, is traced by circles that touch both fixed circles internally — circles that swallow them.

What happens if the two fixed circles have equal radii?

Then PO₁ − PO₂ = 0, so the centre is always equally far from both fixed centres: it moves on the perpendicular bisector of the line joining them. This is a hyperbola that has straightened out completely — its eccentricity has grown without limit.

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