SSC CGL Maths · Circles

Common Tangents and Common Chords

The distance between two centres decides how many common tangents two circles have, and one right triangle with that distance as hypotenuse gives the length of each kind.

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Why this matters

Thirty questions in the bank, 16 EASY, 13 MODERATE and the chapter's only HARD one. Most are a direct or transverse tangent from the radii and the distance between the centres, sometimes run backwards. The MODERATE ones add the two formulas, compare them, or find the chord two intersecting circles share.

Concept 1 of 4: Positions of two circles and the number of common tangents

Pull two circles apart and they gain common tangents; push one inside the other and they lose them. Compare the distance between the centres with the sum and the difference of the radii, and the position, and the count, follow.

Definition

Let $d$ be the distance between the centres and $r_1 \ge r_2$ the radii.
A direct common tangent has both circles on the same side of it. A transverse (internal) one crosses between them.
Circles that cut each other have no transverse tangent: a line between them would have to cross the shared region.
Three circles, each pair outside each other, have at most $3 \times 4 = 12$ common tangents in all.

Position	Condition	Common tangents
One outside the other	$d > r_1 + r_2$	4: two direct, two transverse
Touching externally	$d = r_1 + r_2$	3: two direct, one at the point of contact
Cutting at two points	$r_1 - r_2 < d < r_1 + r_2$	2: both direct
Touching internally	$d = r_1 - r_2$	1: at the point of contact
One inside the other	$d < r_1 - r_2$	0 Concentric circles (d = 0) belong here.

Work out

r_1 + r_2

and

r_1 - r_2

first, then see where

d

falls.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q71Easy

Example 1 · Circles · Common Tangents and Common Chords

Two circles with radii 6 cm and 2 cm have their centers 10 cm apart. How many common tangents can be drawn between them, and what type(s) of tangents are they?

Giving intersecting circles a transverse tangent

Two circles that cut at two points have exactly two common tangents, and both are direct.

Counting three tangents for internal touching

Circles touching externally have 3 common tangents. Circles touching internally have only 1, the tangent at the point of contact.

Drill 4 more on positions of two circles and the number of common tangents

Concept 2 of 4: Length of a direct common tangent

The radii to the two points of contact are both perpendicular to the tangent, so they are parallel. Slide the tangent across by the smaller radius and you get a right triangle: the distance between the centres is the hypotenuse, the tangent is one leg and the difference of the radii is the other.

Definition

Direct common tangent: $L^2 = d^2 - (r_1 - r_2)^2$ .
Run backwards: $d^2 = L^2 + (r_1 - r_2)^2$ , or $(r_1 - r_2)^2 = d^2 - L^2$ .
Equal radii: the difference is 0, so $L = d$ . If the tangent equals the distance between the centres, the radii are equal.
A direct tangent exists whenever neither circle is inside the other, including when they cut each other.
The two direct tangents meet on the line of centres, beyond the smaller circle. That point divides the line of centres externally in the ratio $r_1 : r_2$ , by similar right triangles.
Fast route with surd options: compare squares. $L^2 = 448$ matches $8\sqrt{7}$ because $64 \times 7 = 448$ . An answer must also be less than $d$ .

Direct common tangent

L = \sqrt{d^2 - (r_1 - r_2)^2}

Lthe length of the direct common tangent
dthe distance between the centres
$r_1, r_2$ the two radii

Worked example

Two circles of radii 11 cm and 4 cm have their centres 25 cm apart. Find the length of a direct common tangent.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q71Easy

Example 2 · Circles · Common Tangents and Common Chords

Two circles have radii of 8 cm and 3 cm. The distance between their centers is 15 cm. What is the length of a direct common tangent?

Using the sum of the radii for a direct tangent

The direct tangent uses the difference of the radii, because both radii point the same way from the tangent. The sum belongs to the transverse tangent.

Treating the tangent as the hypotenuse

The distance between the centres is the longest side. A direct common tangent is always shorter than that distance unless the radii are equal.

Drill 10 more on length of a direct common tangent

Concept 3 of 4: Length of a transverse common tangent, and the two tangents together

For a transverse tangent the two radii to the points of contact point in opposite directions. Sliding one across now adds the radii, so the leg of the right triangle is the sum of the radii instead of the difference.

Definition

Transverse common tangent: $M^2 = d^2 - (r_1 + r_2)^2$ . It exists only when the circles are outside each other, $d > r_1 + r_2$ .
Subtract the two formulas: $L^2 - M^2 = (r_1 + r_2)^2 - (r_1 - r_2)^2 = 4r_1 r_2$ . This gives the product of the radii from the two tangents alone.
Add them: $L^2 + M^2 = 2d^2 - 2(r_1^2 + r_2^2)$ .
The transverse tangents cross the line of centres between the centres, dividing it internally in the ratio $r_1 : r_2$ . The two pieces of one transverse tangent, from that point to each point of contact, are in the same ratio.

Transverse common tangent

M = \sqrt{d^2 - (r_1 + r_2)^2} \qquad L^2 - M^2 = 4r_1 r_2

Mthe length of the transverse common tangent
Lthe length of the direct common tangent

Worked example

Two circles of radii 7 cm and 5 cm have their centres 20 cm apart. Find the lengths of a transverse and a direct common tangent.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 1 · Q70Moderate

Example 3 · Circles · Common Tangents and Common Chords

A common internal tangent connecting two circles is 9 cm long. The distance between the circles' centers is 15 cm. If the larger circle has a radius of 8 cm, what is the radius of the smaller circle?

Swapping the sum and the difference

Direct tangent:

d^2 - (r_1 - r_2)^2

. Transverse tangent:

d^2 - (r_1 + r_2)^2

. The transverse one is always the shorter.

Finding a transverse tangent for circles that cut

d

is less than the sum of the radii,

d^2 - (r_1 + r_2)^2

is negative: there is no transverse tangent.

Drill 8 more on length of a transverse common tangent, and the two tangents together

Concept 4 of 4: The common chord of two intersecting circles

Two circles that cut at A and B share the chord AB. Each centre is equally far from A and from B, so both centres lie on the perpendicular bisector of AB. The line of centres therefore cuts the common chord at right angles, at its midpoint.

Definition

The line of centres is the perpendicular bisector of the common chord.
Let the chord's midpoint be $x$ from the first centre and $d - x$ from the second, and let $h$ be half the chord. Then $h^2 = r_1^2 - x^2 = r_2^2 - (d - x)^2$ . Solve the second equality for $x$ ; the $x^2$ terms cancel.
Equal radii: the midpoint is halfway, $x = \frac{d}{2}$ .
Each circle through the other's centre (equal radii $r$ , so $d = r$ ): the common chord is $r\sqrt{3}$ .
The common chord, produced, bisects every common tangent: a point X on line AB has $XP^2 = XA \times XB = XQ^2$ for the points of contact P and Q, by the tangent rule of power of a point.

Common chord

h^2 = r_1^2 - x^2 = r_2^2 - (d - x)^2 \qquad \text{Chord} = 2h

hhalf the common chord
xthe distance from the first centre to the chord's midpoint
dthe distance between the centres

Worked example

Two circles of radii 10 cm and 17 cm cut each other, and their centres are 21 cm apart. Find the length of their common chord.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 24 Jul 2023, Shift 2 · Q54Moderate

Example 4 · Circles · Common Tangents and Common Chords

The length of the common chord of two circles of radii 15 cm and 13 cm, whose centres are 14 cm apart, is:

Putting the chord's midpoint halfway between unequal circles

Only equal circles have the common chord halfway between the centres. For unequal radii, find the split of d from the two right triangles.

Forgetting to double half the chord

The right triangles give half the common chord. Double it before matching the options.

Drill 4 more on the common chord of two 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 (3)

Length of a direct common tangent
Direct common tangent
$L = \sqrt{d^2 - (r_1 - r_2)^2}$
Length of a transverse common tangent, and the two tangents together
Transverse common tangent
$M = \sqrt{d^2 - (r_1 + r_2)^2} \qquad L^2 - M^2 = 4r_1 r_2$
The common chord of two intersecting circles
Common chord
$h^2 = r_1^2 - x^2 = r_2^2 - (d - x)^2 \qquad \text{Chord} = 2h$

Reference tables (1)

Positions of two circles and the number of common tangents5 rows

Position	Condition	Common tangents
One outside the other	$d > r_1 + r_2$	4: two direct, two transverse
Touching externally	$d = r_1 + r_2$	3: two direct, one at the point of contact
Cutting at two points	$r_1 - r_2 < d < r_1 + r_2$	2: both direct
Touching internally	$d = r_1 - r_2$	1: at the point of contact
One inside the other	$d < r_1 - r_2$	0 Concentric circles (d = 0) belong here.

Work out

r_1 + r_2

and

r_1 - r_2

first, then see where

d

falls.

Watch out for (8)

Giving intersecting circles a transverse tangent
→ Positions of two circles and the number of common tangents
Counting three tangents for internal touching
→ Positions of two circles and the number of common tangents
Using the sum of the radii for a direct tangent
→ Length of a direct common tangent
Treating the tangent as the hypotenuse
→ Length of a direct common tangent
Swapping the sum and the difference
→ Length of a transverse common tangent, and the two tangents together
Finding a transverse tangent for circles that cut
→ Length of a transverse common tangent, and the two tangents together
Putting the chord's midpoint halfway between unequal circles
→ The common chord of two intersecting circles
Forgetting to double half the chord
→ The common chord of two intersecting circles

Test yourself on Circles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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