The co-axial system of circles given by ${x^2} + {y^2} + 2gx + c = 0$ for $c < 0$ represents

Circles ${x^2} + {y^2} - 2x - 4y = 0$ and ${x^2} + {y^2} - 8y - 4 = 0$

The circles ${x^2} + {y^2} = 9$ and ${x^2} + {y^2} - 12y + 27 = 0$ touch each other. The equation of their common tangent is 

Radical axis of the circles $3{x^2} + 3{y^2} - 7x + 8y + 11 = 0$ and ${x^2} + {y^2} - 3x - 4y + 5 = 0$ is

The equation of the circle having the lines ${x^2} + 2xy + 3x + 6y = 0$ as its normals and having size just sufficient to contain the circle $x(x - 4) + y(y - 3) = 0$is

Two circles of radii $4$ cms $\&\,\, 1\,\, cm$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $sin \theta =$