The centre of the sphere $\alpha \,r^2 - 2u \cdot r = \beta ,(\alpha \ne 0)$ is

If the plane $2ax - 3ay + 4az + 6 = 0$ passes through the midpoint of the line joining the centres of the spheres ${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$ and ${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$,then $a$ equals

If $A(3, -2, 2)$ and $B(2, \lambda+1, 5)$ are the end points of the diameter of a circle and the point $P(5, 6, -1)$ lies on the circle,then $\lambda=$

Let $(3, 4, -1)$ and $(-1, 2, 3)$ be the end points of a diameter of a sphere. Then the radius of the sphere is equal to

How many distinct spheres of radius $r$ can be drawn such that they touch all three coordinate planes?

$A$ variable plane passes through a fixed point $P(1, 2, 3)$. The foot of the perpendicular from the origin $O(0, 0, 0)$ to the plane lies on: