Find all tangents between circles

Question

Find equation of lines tangent to $(x-4)^2+y^2=4$ and $(x+2)^2+y^2=1$.

I get stuck because I don't know where the lines get tangentially to the circles.

Answer 1

Let the equation be $y=mx+c$. Substitute it into $(x−4)^2+y^2=4$ to obtain a quadratic equation in $x$. This equation has a double root and so its discriminant is $0$. This gives $c$ in terms of $m$.

Put $y=mx+c$ into $(x+2)^2+y^2=1$ to obtain a quadratic equation in $x$, which again has zero discriminant. This gives the equations of tangents.

The tangents are $y=\pm\frac{1}{\sqrt{3}}x$ and $y=\pm\frac{1}{\sqrt{35}}(x+8)$.

Answer 2

Centers of these two circles lie on x axis and they are non-overlapping to each other, so total of 4 tangents will be common between them.

we can assume the equation of tangent to be of type $ y = mx + c $ and as the tangents touch circles solving this equation with circles, along with condition of tangency(perpendicular distance of a tangent from center of circle is equal to radius). will give the values of $m$ and $c$ .