plz solve this problem
Answers
Answer:
If y is the mean proportion between x and z; show that xy+yz is the mean proportional between x^2+ y^2 and y^2+ z^2
Step-by-step explanation:
Since y is the mean proportion between x and z Therefore, y2 = xz Now, we have to prove that xy + yz is the mean proportional between x2 + y2 and y2 + z2, i.e.,
Solution :
If y is the mean proportion between x and z i.e. x:y::y:z
So, y^2=xzy
We have to show, (xy+yz)^2=(x^2+y^2)(y^2+z^2)(xy+yz)
Taking LHS,
(xy+yz)^2=x^2y^2+2xy^2z+y^2z^2(xy+yz)
(xy+yz)^2=x^2xz+2x(xz)z+(xz)z^2(xy+yz)
xz+2x(xz)z+(xz)z
(xy+yz)^2=x^3z+2x^2z^2+xz^3(xy+yz)
(xy+yz)^2=xz(x^2+2xz+z^2)(xy+yz)
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
(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
Answer: