Question

If y is the mean proportion between x and z; show that xy+yz is the mean proportional between x square + y square and y square + z square.

Answer 1

Answer is in the pic below...

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Answer 2

Answer and explanation:

Given : If y is the mean proportion between x and z.

To find : Show that xy+yz is the mean proportional between x square + y square and y square + z square ?

Solution :

If y is the mean proportion between x and z i.e. x:y::y:z

So, $y^2=xz$

We have to show, $(xy+yz)^2=(x^2+y^2)(y^2+z^2)$

Taking LHS,

$(xy+yz)^2=x^2y^2+2xy^2z+y^2z^2$

$(xy+yz)^2=x^2xz+2x(xz)z+(xz)z^2$

$(xy+yz)^2=x^3z+2x^2z^2+xz^3$

$(xy+yz)^2=xz(x^2+2xz+z^2)$

Taking RHS,

$(x^2+y^2)(y^2+z^2)=x^2y^2+x^2z^2+y^4+z^2y^2$

$(x^2+y^2)(y^2+z^2)=x^2(xz)+x^2z^2+(xz)^2+z^2(xz)$

$(x^2+y^2)(y^2+z^2)=x^3z+x^2z^2+x^2z^2+xz^3$

$(x^2+y^2)(y^2+z^2)=x^3z+2x^2z^2+xz^3$

$(x^2+y^2)(y^2+z^2)=xz(x^2+2xz+z^2)$

So, LHS=RHS