Question

The minimum value of $|\left[\mu \right]|$ for which the circle $x^2+y^2=9$ and $x^2+y^2-\mu x-6=0$ have two common tangents is
(where [.] is greatest integer function)

Answer 1

If the two circles have exactly two common tangents, then they will intersect each other at two different points.
$x^2+y^2=9\cdots \left(1\right)$
$x^2+y^2-\mu x-6=0\cdots \left(2\right)$

So, $9-\mu x-6=0$
$\Rightarrow x=\frac{3}{\mu }$

From equation $\left(1\right)$,
$\Rightarrow y=\pm \sqrt{9-{\left(\frac{3}{\mu }\right)}^2}\Rightarrow 9-{\left(\frac{3}{\mu }\right)}^2>0\Rightarrow 9>\frac{9}{{\mu }^2}\Rightarrow {\mu }^2>1\Rightarrow \mu >1\text{or}\mu <-1$

Therefore, minimum value of $|\left[\mu \right]|=1$