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 .