If $4l^2-5m^2+6l+1=0$, then the line $lx+my+1=0$ touches the circle

$4l^2-5m^2+6l+1=0\Rightarrow 4l^2+6l+1=5m^2\Rightarrow 9l^2+6l+1=5\left(l^2+m^2\right)\Rightarrow (3l+1{)}^2=5\left(l^2+m^2\right)$

$\Rightarrow \frac{(3l+1)}{l^2+m^2}=5\Rightarrow \frac{|3l+1|}{\sqrt{l^2+m^2}}=\sqrt{5}\Rightarrow \frac{\left|l\right(3)+m(0)+1|}{\sqrt{l^2+m^2}}=\sqrt{5}$

$\Rightarrow lx+my+1=0$ touches $(x-3{)}^2+y^2=5\Rightarrow lx+my+1=0$ touches $x^2+y^2-6x+4=0$