#### Question

If y is the mean proportional between x and y; show that y(x+z) is the mean p roporti ona I between x^{2}+ y^{2} and y^{2}+ z^{2 }

#### Solution

Since y is the mean proportion between x and z

Therefore, y^{2}=xz

Now, we have to prove that xy+yz is the mean proportional between x^{2}+y^{2} and y^{2}+z^{2}.

(xy + yz)^{2} = (x^{2} + y^{2})(y^{2} + z^{2})

LHS = (xy + yz)^{2}

= [y (x + z)]^{2}

= y^{2}(x + z)^{2}

= xz (x + z)^{2}

RHS = (x^{2} + y^{2})(y^{2} + z^{2})

= (x^{2} + xz) (xz+ z^{2})

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

= xz (x + z)^{2}

LHS = RHS

Is there an error in this question or solution?

Solution If Y is the Mean Proportional Between X and Y; Show that Y(X+Z) is the Mean P Roporti Ona I Between X2+ Y2 and Y2+ Z2 Concept: Proportions.