Sequences And Series

Question
CBSEENMA11015463

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

  • 1

  • 2

  • 3

  • 4

Solution

C.

3

Number of common tangents depend on the position of the circle with respect to the each other.
(i) If circles touch externally ⇒C1C2  = r1+ r2,3 common tangents
(ii) If circles touch internally ⇒ C1C2 = r2-r1, 1 common tangents
(iii) If circles do not touch each other, 4 common tangents
Given  equations 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
=square root of 4 plus 9 plus 12 end root space equals space 5 space left parenthesis straight r subscript 1 right parenthesis
Centre of circle (ii) is C2(-3,-9) and radius
square root of 9 plus 81 minus 26 end root equals space 8 space left parenthesis straight r subscript 2 right parenthesis
Now comma space straight C subscript 1 straight C subscript 2 space equals space square root of left parenthesis 2 plus 3 right parenthesis squared plus left parenthesis 3 plus 9 right parenthesis squared end root
rightwards double arrow space straight C subscript 1 straight C subscript 2 space equals space square root of 5 squared plus 12 squared end root
straight C subscript 1 straight C subscript 2 space equals space square root of 25 plus 144 end root space equals space 13
straight r subscript 1 plus straight r subscript 2 space equals space 5 plus 8 space equals space 13
Also comma space straight C subscript 1 straight C subscript 2 space equals space straight r subscript 1 plus straight r subscript 2
Thus, both circles touch each other externally. Hence, there are three common tangents.