Question

Solve number 9..please

Answer 1

y is the mean proportional between x and z ,
then , y² = xz -----------(1)

(xy + yz) is the mean proportional between (x² + y²) and (y² + z²) only when,
(xy + yz)² = (x² + y²)(y² + z²)

LHS = (xy + yz)² = x²y² + y²z² + 2xy²z
= x²y² + y²z² + 2(xz).y²
= x²y² + y²z² + 2(xz)(xz) [ put y² = xz from (1)
=x²y² + y²z² + x²z² + x²z²
=(x²y² + x²z² ) + (xy)² + y²z²
=x²(y² + z²) + y⁴ + y²z²
=x²(y² + z²) + y²(y² + z²)
= (y² + z²)(x² + y²) = RHS