The locus of the centre of the circle which cuts a chord of length $2a$ from the positive $x$-axis and passes through a point on the positive $y$-axis at a distance $b$ from the origin is:

If $A=(1,2), B=(2,1)$ and $P$ is a variable point satisfying the condition $|PA-PB|=3$,then the locus of $P$ is

If a tangent to the circle $x^2 + y^2 = 1$ intersects the coordinate axes at distinct points $P$ and $Q,$ then the locus of the mid-point of $PQ$ is

If $A (c, 0)$ and $B (-c, 0)$ are two points,find the locus of point $P$ such that $PA^{2} + PB^{2} = AB^{2}$.

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 . . . . . .

If $A(2, 3)$ and $B(3, -2)$ are two fixed points and $P(x, y)$ is a variable point satisfying the condition $|PA - PB| = 2$,then the locus of $P$ is