Question

If $x^2+4y^2=12xy,x\in [1,4],y\in [1,4],$ then which of the following condition is true?

• A. the greatest value of ${\log }_2(x+2y)$ is $5$
• B. the least value of ${\log }_2(x+2y)$ is $3$
• C. the range of values of ${\log }_2(x+2y)$ lies in $(2,4)$
• D. the number of integral values of $(x,y)$ are 3 such that ${\log }_2(x+2y)$ is equal to $3$

Answer 1

Answer: (C)

the range of values of ${\log }_2(x+2y)$ lies in $(2,4)$

$x^2+4y^2=12xy$
$\Rightarrow (x+2y{)}^2=16xy$
Taking logarithm with base 2 on both sides
$\Rightarrow 2{\log }_2(x+2y)={\log }_{2}16+{\log }_{2}xy$
$\Rightarrow 2{\log }_2(x+2y)=4+{\log }_{2}xy$ .....(1)
Since, $x,y\in [1,4]$
$\Rightarrow 1\le xy\le 16$
$\Rightarrow {\log }_{2}1\le {\log }_2\left(xy\right)\le {\log }_2\left(16\right)$
$\Rightarrow 0\le {\log }_2\left(xy\right)\le 4$
$\Rightarrow 4\le 4+{\log }_2\left(xy\right)\le 8$
$\Rightarrow 4\le 2{\log }_2(x+2y)\le 8$ (by (1))
$\Rightarrow 2\le {\log }_2(x+2y)\le 4$
Hence, the range of ${\log }_2(x+2y)$ is $[2,4]$