Tangency of a Plane to a Sphere
The plane lx + my + nz = p will touch the sphere x 2 + y 2 + z 2 + 2ux + 2vy + 2 wz + d = 0, if length of the perpendicular from the centre( – u, – v,— w)= radius,
i. e., | lu – mv – nw – p | / √l 2 + m 2 + n 2 = √u 2 + v 2 + w 2 – d( lu – mv – nw – p) 2 =( u 2 + v 2 + w 2 – d)( l 2 + m 2 + n 2)
Plane Section of a Sphere
Consider a sphere intersected by a plane. The set of points common to both sphere and plane is called a plane section of a sphere.
In ΔCNP, NP 2 = CP 2 – CN 2 = r 2 – p 2 ∴ NP = √r 2 – p 2
Hence, the locus of P is a circle whose centre is at the point N, the foot of the perpendicular from the centre of the sphere to the plane.
The section of sphere by a plane through its centre is called a great circle. The centre and radius of a great circle are the same as those of the sphere.