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

If $A$ and $B$ are the points of intersection of the circles $x^2+y^2-4x+6y-3=0$ and $x^2+y^2+2x-2y-2=0$,then the distance between $A$ and $B$ is

$C_1$ is the circle with centre at $O(0,0)$ and radius $4$,$C_2$ is a variable circle with centre at $(\alpha, \beta)$ and radius $5$. If the common chord of $C_1$ and $C_2$ has slope $\frac{3}{4}$ and is of maximum length,then one of the possible values of $\alpha+\beta$ is

Two circles $S_1 = px^2 + py^2 + 2g'x + 2f'y + d = 0$ and $S_2 = x^2 + y^2 + 2gx + 2fy + d' = 0$ have a common chord $PQ$. The equation of $PQ$ 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:

What is the angle subtended by the common chord of the circles $x^2 + y^2 - 4x - 4y = 0$ and $x^2 + y^2 = 16$ at the origin?