The length of the common chord of the circles $x^2+y^2+3x+5y+4=0$ and $x^2+y^2+5x+3y+4=0$ is

The length of the chord of contact of tangents drawn from the point $(4,4)$ to the circle $x^2 + y^2 - 2x - 2y - 7 = 0$ is: (in $\sqrt{2}$)

If the chord of contact of the point $P(h, k)$ with respect to the circle $x^2+y^2-4x-4y+8=0$ meets the circle in two distinct points and it also makes an angle $45^{\circ}$ with the positive $X$-axis in the positive direction,then $(h, k)$ cannot be

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

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2+4x+6y+4=0$ is

$A$ circle $C_1$ of unit radius lies in the first quadrant and touches both coordinate axes. The radius of another circle that touches both coordinate axes and intersects $C_1$ such that the common chord is of maximum length is: