The number of common tangents that can be drawn to the circles x^2+y^2-6 x=0 and x^2+y^2+6 x+2 y+1=0 is .....

Step-by-step solution

Common tangents to two circles The centers and radii of the circles are determined using the standard circle equation x^2+y^2+2gx+2fy+c=0 . For the first circle x^2+y^2-6x=0 , the center is C₁=(3,0) and radius r₁=3 . For the second circle x^2+y^2+6x+2y+1=0 , the center is C₂=(-3,-1) and radius r₂=3 . The distance between centers is d= (-3-3)^2+(-1-0)^2 = 37 . Since d>r₁+r₂ because 37 >6 , the circles lie completely outside each other. Two non-intersecting circles have four common tangents: two direct and two transv