The common chord of the circles $x^2 + y^2 + 4x + 1 = 0$ and $x^2 + y^2 + 6x + 2y + 3 = 0$ is

If the chord of contact of tangents from a point $A$ to a given circle passes through $B$,then the circle with $AB$ as a diameter will . . . . . .

The length of the common chord of the circles $x^2 + y^2 + 2x + 4y - 20 = 0$ and $x^2 + y^2 + 6x - 8y + 10 = 0$ is

The length of the common chord of the circles ${x^2} + {y^2} + 5x + 7y + 9 = 0$ and ${x^2} + {y^2} + 7x + 5y + 9 = 0$ is

The circle $S \equiv x^2+y^2-2x-4y+1=0$ cuts the $y$-axis at $A, B$ $(OA > OB)$. If the radical axis of $S=0$ and $S^{\prime} \equiv x^2+y^2-4x-2y+4=0$ cuts the $y$-axis at $C$,then the ratio in which $C$ divides $AB$ is:

Tangents are drawn from any point on the hyperbola $4x^2 - 9y^2 = 36$ to the circle $x^2 + y^2 = 9$. If the locus of the midpoint of the chord of contact is $\left( \frac{x^2}{9} - \frac{y^2}{4} \right) = \lambda \left( \frac{x^2 + y^2}{9} \right)^2$,then the value of $\lambda$ is: