The number of common tangents to the circles `"x"^("2")"+y"^("2")"-4x-6y-12=0"` and `x^2+""y^2+""6x""+""18 y""+""26""=""0` , is :

Options

A. 1
B. 2
C. 3
D. 4

Answer

C

Explanation

PLAN Number of common tangents depend on the position of the circle with respect to each other.
(i) If circles touch internally `rArr C_(1)C_(2)=r_(1)+r_(2)`, 3 common tangents
(ii) If circles touch internally `rArr C_(1)C_(2)=r_(1)+r_(2)`, 1 common tangent.
(iii) If circles do not touch each other, 4 common tangents.
Given equations of circles are
`x^(2)+y^(2)-4x -6y-12=0" "...(i)`
`x^(2)+y^(2)+6x+18y + 26=0" " ...(ii)`
Centre of circle (i) is `C_(1)(2, 3)` and radius
`=sqrt(4+9+12)=5(r_(1))" "["say"]`
Centre of circle (ii) is `C_(2)(-3, -9)` and radius
`=sqrt(9+81-26)=8(r_(2))" " ["say"]`
Now, `C_(1)C_(2)=sqrt((2+3)^(2)+(3+9)^(2))`
`rArr C_(1)C_(2)=sqrt(5^(2)+12^(2))`
`rArrC_(1)C_(2)=sqrt(25+144)=13`
`thereforer_(1
)+r_(2)=5+8+13`
Also `C_(1)C_(2)=r_(1)+r_(2)`
Thus, both circles touch each other externally, Hence, there are three common tangents.