Question

How many common tangents can be drawn to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$.
What is the nature of the figure formed by these tangents?

Answer 1

$C_1(3,0),r_1=3;C_2(-1,0),r_2=1$
Distance between centres is $4=r_1+r_2$. Hence they touch and common tangent by $S_1-S_2=0$ is $x=0$ i.e. y - axis. Other two direct common tangents are found as in to be $x-\sqrt{3}y+3=0$ and $x+\sqrt{3}y+3=0$ being drawn from the point $(-3,0)$. These two tangents meet the common tangent $x=0$ at $B(0,\sqrt{3})$ and $C(0,-\sqrt{3})$. Clearly $ABC$ is equilateral as each side $=2\sqrt{3}$. Hence the figure formed by these tangents is an equilateral triangle.