Question

If y is the mean proportional between x and z; show that xy + yz is the mean proportional between x² + y² and y² + z².

Answer 1

Since y is the mean proportion between x and z

Therefore, y² = xz

Now, we have to prove that xy + yz is the mean proportional between x² + y² and y² + z²,

i.e., (xy + yz)² = (x² + y²)(y² + z²)

LHS = (xy + yz)²

= [y(x + z)]²

= y²(x + z)²

= xz(x + z)²

RHS = (x² + y²)(y² + z²)

= (x² + xz)(xz + z²)

= x(x + z) z (x + z)

= xz(x + z)²

LHS = RHS

Hence, proved.