How do I show that the line $lx+my+1=0$ touches a fixed circle, provided $l^2-5m^2+6l+1=0$?

Question

A high school geometry problem.

How do I show that if the line $lx+my+1=0$ touches a fixed circle, provided $4l^2-5m^2+6l+1=0$?

Answer 1

Hint:

You have to find such point $S(a,b)$ and $r>0$ that $${|la+mb+1|\over \sqrt{m^2+l^2}} =r$$

To do that think of three diffrent pair of $l,m$ in orther to get 3 lines and calculate their intesection points. Then calculate the radius of it incircle and coordinates of incenter.

But, as far as I can see three is no such circle. If we say $m=0$ we get two lines parallel to $y$ axsis and if we take $l=0$ we get two lines parallel to $x$ axsis. These lines form rectangle which is not a square so there is no such circle.