Question

If x^2+ 5y^2+ z^2 = 2y (2x + z), then which of the following statements are necessarily true?
I. x = 2y II. x = 2z III. 2x = z
(1) Only I
(2) Only II
(3) Only III
(4) Only I and II

Answer 1

x^2+5y^2+z^2=2y(2x+z)
Or, x^2+4y^2+y^2+z^2+=4xy+2yz
Or, x^2-4xy+4y^2+y^2-2yz+z^2=0
Or, (x-2y)^2+(y-z)^2=0
hence (x-2y)^2=0
Or, x-2y=0
Or, x=2y
And (y-z)^2=0
Or, y=z
therefore answer is (1) only |