External and Internal Contacts of Circles

If two circles with centres C₁(x₁, y₁) and C₂(x₂, y₂) and radii r₁ and r₂ respectively, touch each other externally, C₁C₂ = r₁ + r₂. Coordinates of the point of contact are.

The circles touch each other internally if

C₁C₂ = r₁- r₂.

Coordinates of the point of contact are

Common Tangents to Two Circles :

(a) The direct common tangents to two circles meet on the line of centres and divide it externally in the ratio of the radii.

(b) The transverse common tangents also meet on the line of centres and divide it internally in the ratio of the radii.

Note :

When two circles are real and non-intersecting, 4 common tangents can be drawn.
When two circles touch each other externally, 3 common tangents can be drawn to
the circles.
When two circles intersect each other, two common tangents can be drawn to the circles. .
When two circles touch each other internally 1 common tangent can be drawn to
the circles.

Example 1

Examine whether the two circles x²+y²-2x-4y = 0 and x²+y²-8y-4=0 touch each other externally or internally.

Solution:

Let C₁ and C₂ be the centres of the circles.

&⇒ C₁ º(1, 2) and C₂ º (0, 4). Let r₁ and r₂ be the radii of the circles.

&⇒ r₁ = Ö5 and r₂ = 2Ö5

Also C₁C₂ = =

But r₁ + r₂ = 3Ö5 and r₂ - r₁ = Ö5 = C₁C₂.

Hence the circles touch each other internally.

Example 2

Prove that x² + y² = a² and (x – 2a)² + y² = a² are two equal circles touching each other. Find the equation of circle (or circles) of the same radius touching both the circles.

Solution:

Given circles are

x² + y² = a² …(1)

and (x – 2a)² + y² = a²…(2)

Let A and B be the centres and r₁ and r₂ the radii of the two circles (1) and (2) respectively. Then.

A º (0, 0), B º (2a, 0), r₁ = a, r₂ = a

Now AB =

Hence the two circles touch each other externally.

Let the equation of the circle having same radius ‘a’ and touching the circles (1) and (2) be

(x – a)² + (y – b)² = a² …(3)

Its centre C is (a, b) and radius r₃ = a

Since circle (3) touches the circle (1),

AC = r₁ + r₃ = 2a. [Here AC > |r₁ – r₃| as r₁– r₃= a – a = 0].

&⇒ AC² = 4a²
&⇒ a² + b² = 4a² …(4)

Again since circle (3) touches the circle (2)

BC = r₂ + r₃
&⇒ BC² = (r₂ + r₃)²

&⇒ (2a – a)² + b² = (a + a)²
&⇒ a² + b² – 4a a = 0

&⇒ 4a² – 4a a = 0 [from (4)]

&⇒ a = a and from (4), we have b = ± a.

Hence, the required circles are

(x – a)² + ( y aÖ3)² = a²

or x² + y² – 2ax 2Ö3ay + 3a² = 0

External and Internal Contact of Circles

External and Internal Contacts of Circles

Common Tangents to Two Circles :

Example 1

Solution:

Example 2

Solution:

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