Question

If Y is the mean proportional between X and Z, find the mean proportional between X square + Y square and Y square and Z square

Answer 1

Y is the mean proportion of X and Z
so, Y = √XZ
e.g Y²= ZX ------(1)

now , mean proportion between ( X² + Y²) and ( Y² + Z²)
= √(X² + Y²)(Y² + Z²)
=√(X²Y² +X²Z² + Y⁴ + Y²Z²)
=√(X²Y² + Y⁴ + Y⁴ + Y²Z²) from eqn (1)
=√( 2Y⁴ + X²Y² +Y²Z²)
=Y√(2Y² + X² + Z²)
=Y√(2ZX + X² + Z²) from eqn (1)
=Y√(X + Z)²
=Y(X + Z)