#### Question

If y is the mean proportional between x and z; show that xy + yz is the mean proportional 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}, i.e., `(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

Hence, proved.

Is there an error in this question or solution?

Solution If Y is the Mean Proportional Between X and Z; Show that Xy + Yz is the Mean Proportional Between X2+Y2 And Y2+ Z2. Concept: Proportions.