MHT-CET Maths · Circle

Two Circles — Touching, Common Tangents and Relative Position

Compare the distance d between the centres with r₁ + r₂ and |r₁ − r₂|: d > r₁ + r₂ gives 4 common tangents, d = r₁ + r₂ external touching and 3, |r₁ − r₂| < d < r₁ + r₂ cutting and 2, d = |r₁ − r₂| internal touching and 1, d < |r₁ − r₂| one inside the other and 0.

All Circle notes MHT-CET Maths strategy

Why this matters

9 PYQs at 44% HARD. Of the seven set from 2021 to 2025, four were the count of common tangents (the 2025 stem and three earlier), one was the external-touching condition for x² + y² + 2ax + c = 0 and x² + y² + 2by + c = 0 (2024), one was internal touching with a parameter, and one asked for the centre of a circle touching a given circle internally at a given point. The whole page is one comparison — d against the sum and the difference of the radii.

Concept 1 of 2: Relative Position From d, r₁ + r₂ and |r₁ − r₂|: How Many Common Tangents

Compute the two centres and radii, the distance

d

between the centres, and place

d

against

r_1 + r_2

and

|r_1 - r_2|

. Each of the five positions has its own number of common tangents:

4, 3, 2, 1, 0

as the circles move from apart to nested.

Definition

$x^2 + y^2 - 6x - 14y + 48 = 0$ ( $(3, 7)$ , $\sqrt{10}$ ) and $x^2 + y^2 - 6x = 0$ ( $(3, 0)$ , $3$ ): $d = 7 > 3 + \sqrt{10}$ : apart, $4$ tangents.
$x^2 + y^2 - 6x = 0$ ( $(3, 0)$ , $3$ ) and $x^2 + y^2 + 6x + 2y + 1 = 0$ ( $(-3, -1)$ , $3$ ): $d = \sqrt{37} > 6$ : $4$ .
$x^2 + y^2 - x = 0$ and $x^2 + y^2 + x = 0$ : centres $\left(\pm\tfrac12, 0\right)$ , radii $\tfrac12$ ; $d = 1 = r_1 + r_2$ : touch externally, $3$ .
$x^2 + y^2 + 6x + 6y = 0$ ( $(-3, -3)$ , $3\sqrt2$ ) and $x^2 + y^2 - 12x - 12y = 0$ ( $(6, 6)$ , $6\sqrt2$ ): $d = 9\sqrt2 = r_1 + r_2$ : touch externally.
Orthogonal circles ( $d^2 = r_1^2 + r_2^2$ , i.e. $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ ) are a special case of cutting.

Five positions

d > r_1 + r_2:\ 4 \quad d = r_1 + r_2:\ 3 \quad |r_1 - r_2| < d < r_1 + r_2:\ 2 \quad d = |r_1 - r_2|:\ 1 \quad d < |r_1 - r_2|:\ 0

Worked example

How many common tangents do

x^2 + y^2 = 4

and

x^2 + y^2 - 8x + 12 = 0

have?

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 11th May Shift 1 · Q106Moderate

Example 1 · Circle · Two Circles — Touching, Common Tangents and Relative Position

Number of common tangents to the circles

x^2+y^2-6x-14y+48=0

and

x^2+y^2-6x=0

are

Comparing d with r₁ + r₂ only

d < r_1 + r_2

covers three positions (cutting, internal touch, nested). The second comparison, against

|r_1 - r_2|

, decides between

2

1

and

0

Drill 3 more on relative position from d, r₁ + r₂ and |r₁ − r₂|: how many common tangents

Concept 2 of 2: Touching Circles: d = r₁ + r₂ (External) or d = |r₁ − r₂| (Internal), and the Centre From the Contact Point

Touching is an equation, not an inequality, so it fixes a parameter. Square carefully — the radii carry square roots. When the contact point is known, the centres and the contact point are collinear, and the section formula places the new centre.

Definition

$x^2 + y^2 + 2ax + c = 0$ ( $(-a, 0)$ , $\sqrt{a^2 - c}$ ) and $x^2 + y^2 + 2by + c = 0$ ( $(0, -b)$ , $\sqrt{b^2 - c}$ ) touching externally: $\sqrt{a^2 + b^2} = \sqrt{a^2 - c} + \sqrt{b^2 - c}$ ; squaring twice: $c^2 = (a^2 - c)(b^2 - c) \Rightarrow c(a^2 + b^2) = a^2b^2 \Rightarrow \dfrac{1}{a^2} + \dfrac{1}{b^2} = \dfrac1c$ .
$x^2 + y^2 = 9$ and $x^2 + y^2 + 2\alpha x + 2y + 1 = 0$ ( $(-\alpha, -1)$ , $r = |\alpha|$ ) touching internally: $\sqrt{\alpha^2 + 1} = 3 - |\alpha| \Rightarrow \alpha^2 + 1 = 9 - 6|\alpha| + \alpha^2 \Rightarrow |\alpha| = \tfrac43$ ; $\alpha^3 = \tfrac{64}{27}$ .
Circle of radius $3$ touching $x^2 + y^2 - 4x - 6y - 12 = 0$ ( $(2, 3)$ , $r = 5$ ) internally at $(-1, -1)$ : the new centre is on the segment from $(2, 3)$ to $(-1, -1)$ , at distance $5 - 3 = 2$ from the big centre, i.e. dividing it $2 : 3$ : $\left(\dfrac{-2 + 6}{5}, \dfrac{-2 + 9}{5}\right) = \left(\dfrac45, \dfrac75\right)$ .
For internal touching the smaller centre lies BETWEEN the larger centre and the contact point; for external touching the contact point lies between the two centres.

Touching

\text{external: } d = r_1 + r_2 \qquad \text{internal: } d = |r_1 - r_2| \qquad \text{centres and contact point are collinear}

Worked example

Find

k

x^2 + y^2 = 4

and

(x - 5)^2 + y^2 = k^2

(

k > 0

) touch externally.

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 12th May Shift 2 · Q148Hard

Example 2 · Circle · Two Circles — Touching, Common Tangents and Relative Position

The circles

x^2+y^2+2ax+c=0

and

x^2+y^2+2by+c=0

touch each other externally, if

Squaring once and stopping

\sqrt{a^2 + b^2} = \sqrt{a^2 - c} + \sqrt{b^2 - c}

needs TWO squarings; after the first,

2c = 2\sqrt{(a^2 - c)(b^2 - c)}

still has a root. The half-done version gives

\dfrac{1}{a^2} + \dfrac{1}{b^2} = \dfrac{1}{c^2}

, option (C).

Drill 2 more on touching circles: d = r₁ + r₂ (external) or d = |r₁ − r₂| (internal), and the centre from the contact point

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)

Relative Position From d, r₁ + r₂ and |r₁ − r₂|: How Many Common Tangents
Five positions
$d > r_1 + r_2:\ 4 \quad d = r_1 + r_2:\ 3 \quad |r_1 - r_2| < d < r_1 + r_2:\ 2 \quad d = |r_1 - r_2|:\ 1 \quad d < |r_1 - r_2|:\ 0$
Touching Circles: d = r₁ + r₂ (External) or d = |r₁ − r₂| (Internal), and the Centre From the Contact Point
Touching
$\text{external: } d = r_1 + r_2 \qquad \text{internal: } d = |r_1 - r_2| \qquad \text{centres and contact point are collinear}$

Watch out for (2)

Comparing d with r₁ + r₂ only
→ Relative Position From d, r₁ + r₂ and |r₁ − r₂|: How Many Common Tangents
Squaring once and stopping
→ Touching Circles: d = r₁ + r₂ (External) or d = |r₁ − r₂| (Internal), and the Centre From the Contact Point

Test yourself on Circle

15 past MHT-CET questions from this chapter, timed at 27 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 Circle chapter test Drill all 9 questions

Test yourself on Circle

Related notes