Question

If $x^2+5y^2+z^2=2y(2x+z)$, then which of the following statement(s) is/are necessarily true?
(i) $x=2y$ (ii) $y=z$ (iii) $2x=z$

• A. only (i)
• B. only (i) and (iii)
• C. only (i) and (ii)
• D. only (ii)

Answer 1

Answer: (C)

only (i) and (ii)

$x^2+5y^2+z^2=4yx+2yz$
$\Rightarrow (x^2+4y^2-4yx)+(z^2+y^2-2yz)=0$
$\Rightarrow (x-2y{)}^2+(z-y{)}^2=0$
This expression can be true only if $x=2y$ and $2z=x.$