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CDS Mathematics · Circles

Two Circles: Common Tangents and Common Chords

A common tangent's length comes from a right triangle on the line of centres; a common chord is bisected at right angles by that line.

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Why this matters

Six PYQs. Every one draws the line joining the centres. For a tangent, slide it parallel until it passes through a centre: the right triangle has hypotenuse d and one leg the difference (direct) or sum (transverse) of the radii. For a chord, the line of centres is its perpendicular bisector.

Concept 1 of 2: Direct and transverse common tangents

The radii to a common tangent are both perpendicular to it, so they are parallel. Shift the tangent until it passes through the smaller circle's centre: what is left is a right triangle with the distance between the centres as hypotenuse.

Definition

  • Direct common tangent (both circles on one side): d2−(r1−r2)2\sqrt{d^2 - (r_1 - r_2)^2}.
  • Transverse common tangent (crosses between them): d2−(r1+r2)2\sqrt{d^2 - (r_1 + r_2)^2}, which exists only when d>r1+r2d > r_1 + r_2.
  • Circles touching externally: d=r1+r2d = r_1 + r_2, so the direct tangent is 2r1r22\sqrt{r_1 r_2}.
  • The radii and the direct tangent form a right trapezium with parallel sides r1,r2r_1, r_2 and height the tangent length.
  • Common tangents: 44 if apart, 33 if touching externally, 22 if intersecting, 11 if touching internally, 00 if one is inside the other.

Common tangents

direct=d2−(r1−r2)2,transverse=d2−(r1+r2)2\text{direct} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad \text{transverse} = \sqrt{d^2 - (r_1 + r_2)^2}

Worked example

Circles of radii 1010 cm and 44 cm have centres 1010 cm apart. Find the direct common tangent.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q66Moderate

Example 1 · Circles · Common Tangents and Common Chords

The distance between the centres of two circles having radii 9 cm and 4 cm is 13 cm. What is the length of the direct common tangent of these circles ?

Difference for direct, sum for transverse

The direct tangent keeps both circles on one side, so the leg is the DIFFERENCE of the radii. The transverse tangent crosses between them and uses the SUM. Swapping them is the standard wrong option.
Drill 3 more on direct and transverse common tangents

Concept 2 of 2: The common chord of intersecting circles

Both centres are equally far from the chord's two ends, so both lie on its perpendicular bisector. The line of centres cuts the chord in half at right angles, and each centre gives a right triangle.

Definition

  • The line of centres is the perpendicular bisector of the common chord.
  • With half-chord hh: each centre is r2−h2\sqrt{r^2 - h^2} from the chord.
  • Centres on opposite sides of the chord: d=r12−h2+r22−h2d = \sqrt{r_1^2 - h^2} + \sqrt{r_2^2 - h^2}. Same side: the difference.
  • Equal circles each through the other's centre: d=rd = r and the common chord is r3r\sqrt3.

Distance between centres

d=r12−h2+r22−h2d = \sqrt{r_1^2 - h^2} + \sqrt{r_2^2 - h^2}

Worked example

Circles of radii 1515 cm and 1313 cm have a common chord 2424 cm long. Find the distance between their centres.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q87Moderate

Example 2 · Circles · Common Tangents and Common Chords

Two equal circles intersect such that each passes through the centre of the other. If the length of the common chord of the circles is 10310\sqrt{3} cm, then what is the diameter of the circle ?

Both centres can be on one side

When the smaller circle's centre lies on the same side of the chord as the larger one's, dd is the DIFFERENCE of the two distances. Questions mean the usual picture, centres on opposite sides, unless they say otherwise.
Drill 1 more on the common chord of intersecting circles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Direct and transverse common tangents

    Common tangents

    direct=d2−(r1−r2)2,transverse=d2−(r1+r2)2\text{direct} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad \text{transverse} = \sqrt{d^2 - (r_1 + r_2)^2}
  • The common chord of intersecting circles

    Distance between centres

    d=r12−h2+r22−h2d = \sqrt{r_1^2 - h^2} + \sqrt{r_2^2 - h^2}

Watch out for (2)

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter testDrill all 6 questions