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

Given,

y is the mean proportional between x and z

$\therefore \dfrac{x}{y} = \dfrac{y}{z} \Rightarrow y^2 = xz.$

To prove,

xy + yz is the mean proportional between x² + y² and y² + z²

$\therefore \dfrac{x^2 + y^2}{xy + yz} = \dfrac{xy + yz}{y^2 + z^2} \\[1em] \Rightarrow (xy + yz)(xy + yz) = (x^2 + y^2)(y^2 + z^2) \\[1em] \Rightarrow x^2y^2 + xy^2z + xy^2z + y^2z^2 = x^2y^2 + x^2z^2 + y^4 + y^2z^2 ….[\text{i}]$

Substituting y² = xz in L.H.S. of (i)

⇒ x²(xz) + x(xz)z + x(xz)z + (xz)z²

⇒ x³z + x²z² + x²z² + xz³

⇒ x³z + 2x²z² + xz³ ………(ii)

Substituting y² = xz in R.H.S. of (i)

⇒ x²(xz) + x²z² + (xz)² + (xz)z²

⇒ x³z + x²z² + x²z² + xz³

⇒ x³z + 2x²z² + xz³ ………(iii)

Since, (ii) = (iii)

Hence, proved that xy + yz is the mean proportional between x² + y² and y² + z².