The locus of the center of the circle which cuts the circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ orthogonally is

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at points $A$ and $B$. Then the minimum value of $OA+OB$,where $O$ is the origin,is equal to

The locus of the point whose ratio of distance from the origin to its distance from $(-2, -3)$ is $5: 7$,is given by:

For two points $A(2,1)$ and $B(1,2)$,$P$ is a point such that $PA:PB = 2:1$. Then the locus of $P$ is:

The number of pairs $(a, b)$ of positive real numbers satisfying $a^4+b^4 < 1$ and $a^2+b^2 > 1$ is

Consider the circle $C : x^2 + y^2 - 6x - 8y - 11 = 0$. Let a variable chord $AB$ of the circle $C$ subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord $AB$ is the circle $x^2 + y^2 - \alpha x - \beta y - \gamma = 0$, then $\alpha + \beta + 2\gamma$ is equal to . . . . . .