Question

The number of common tangents to the circles x² + y² – 4x – 6x – 12 = 0 and x² + y² + 6x + 18y + 26 = 0 is

पर्याय

• 3

• 4

• 1

• 2

Answer 1

3

Explanation:

The number of common tangents depends on the position of the circles in relation to each other.

(i) If circle touch externally

⇒ C₁C₂ = r₁ + r₂, 3 common tangents

(ii) If circles touch internally

⇒ C₁C₂ = r₂ – r₁, 1 common tangent

(iii) If circles do not touch each other, 4 common tangents

Given equation of circles are

x² + y² – 4x – 6y – 12 = 0  ......(i)

x² + y² + 6x + 18y + 26 = 0 ......(ii)

Centre of circle (i) is C₁(2, 3) and radius

= `sqrt(4 + 9 + 12)` = 5(r₁)  .....(say)

Centre of circle (ii) is C₂(–3, – 9) and radius

= `sqrt(9 + 81 - 26)` = 8(r₂)  .....(say)

Now, C₁C₂ = `sqrt((2 + 3)^2 + (3 + 9)^2)`

⇒ C₁C₂ = `sqrt(5^2 + 12^2)`

⇒ C₁C₂ = `sqrt(25 + 144)` = 13

∴ r₁ + r₂ = 5 + 8 = 13

Also, C₁C₂ = r₁ + r₂

As a result, both circles are externally touching.

As a result, there are three common tangents.