Question

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 : (1) 1 (2) 2 (3) 3 (4) 4

Answer 1

`x^2 + y^2 - 4x - 6y - 12= 0`
`x^2 - 4x + 4 + y^2 - 6y+ 9 - 12 - 4 - 9 = 0`
`(x-2)^2 + (y-3)^2 = 25= 5^2`
`c_1 = (2,3) & r_1= 5`
`x^2 + y^2 + 6x + 18y + 26= 0`
`x^2 + 6x + 9 + y^2 + 18y + 81 + 26 - 9 - 81= 0`
`(x+3)^2 + (y+9)^2 = 64= 8^2`
`c_2 = (-3,-9) & r_2= 8`
`c_1 = (2,3); r_1=5`
`c_2 = (-3,-9) ; r_2= 8`
`d(c_1,c_2) = sqrt((2- (-3))^2 + (3- (-9))^2`
`= sqrt(25 + 144) = sqrt(169)= 13`
`r_1 = 5 ; r_2= 8`
`r_1 + r_2 = 13`
so,`3 `tangents
option 2 is correct