Circles Test 14

Equation of a circle is  $$S={x}^{2}+{y}^{2}+2gx+2fy+c$$
Its notation is $${S}_{1}={x}_{1}x+{y}_{1}y+g\left(x+{x}_{1}\right)+f\left({y}_{1}+y\right)+c$$
$${S}_{11}={x}_{1}^{2}+{y}_{1}^{2}+2g{x}_{1}+2f{y}_{1}+c$$
$$\left(i\right)$$Location of $$P\left({x}_{1},{y}_{1}\right):$$
$$P$$ lies inside the circle $$S=0$$, if $${S}_{11}<0$$
$$P$$ lies outside the circle $$S=0$$ if $${S}_{11}>0$$
$$P$$ lies on the circle $$S=0,$$ if $${S}_{11}=0$$
$$\left(ii\right)$$Tangent at $$P\left({x}_{1},{y}_{1}\right)$$ on the circle $$S=0,$$ is $${S}_{1}=0$$
$$\left(iii\right)$$ Length of the tangent from the point $$\left({x}_{1},{y}_{1}\right)$$ to the circle $$S=0$$ is $$\sqrt{{S}_{11}}$$
$$\left(iv\right)$$Pair of tangents $$PQ,PR$$ from $$P\left({x}_{1},{y}_{1}\right)$$ is $${S}_{1}^{2}={S}_{11}S$$
$$\left(v\right)$$Chord of contact $$QR$$ of tangents from $$P\left({x}_{1},{y}_{1}\right)$$ is $${S}_{1}=0$$
$$\left(vi\right)$$Chord of $$S=0$$ with midpoint $$\left({x}_{1},{y}_{1}\right)$$ is $${S}_{1}={S}_{11}$$
Based on the above information,answer the following questions:

If $$7{l}^{2}-9{m}^{2}+8l+1=0$$ and we have to find the equation of circle having $$lx+my+1=0$$ is a tangent and we can adjust given condition as $$16{l}^{2}+8l+1=9\left({l}^{2}+{m}^{2}\right)$$
or $${\left(4l+1\right)}^{2}=9\left({l}^{2}+{m}^{2}\right)\Rightarrow \dfrac{\left|4l+1\right|}{\sqrt{\left({l}^{2}+{m}^{2}\right)}}=3$$
Center of circle$$=\left(4,0\right)$$ and radius$$=3$$ when any two non-parallel lines touching a circle, then centre of circle lies on angle bisector of lines.