Question

Assertion :If $x,y,z$ are positive and are connected by the relations:

${\log }_{3}x+{\log }_{9}y+{\log }_{9}z=2$

${\log }_{4}x+{\log }_{2}y+{\log }_{4}z=2$

${\log }_{25}x+{\log }_{25}y+{\log }_{5}z=2$,

then numerical value of $yz+zx+xy$ is $\frac{3731}{900}$

Reason: If $x,y$ are positive and $a,b>0,a,b\ne 1$, then
${\log }_a\left(xy\right)={\log }_{a}x+{\log }_{a}y$ and ${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Reason is true (properties of logarithm)
Assertion:
${\log }_{3}x+{\log }_{9}y+{\log }_{9}z=2$
$\Rightarrow {\log }_{3}x+\frac{1}{2}{\log }_{3}y+\frac{1}{2}{\log }_{3}z=2\Rightarrow {\log }_{3}x\sqrt{yz}=2\Rightarrow x\sqrt{yz}=3^2=9{\log }_{4}x+{\log }_{2}y+{\log }_{4}z=2$
$\Rightarrow \frac{1}{2}{\log }_{2}x+{\log }_{2}y+\frac{1}{2}{\log }_{2}z=2\Rightarrow {\log }_2\sqrt{xz}y=2\Rightarrow \sqrt{xz}y=2^2=4{\log }_{25}x+{\log }_{25}y+{\log }_{5}z=2$
$\Rightarrow \frac{1}{2}{\log }_{5}x+\frac{1}{2}{\log }_{5}y+{\log }_{5}z=2\Rightarrow {\log }_5\sqrt{xy}z=2\Rightarrow \sqrt{xy}z=5^2=25$
Therefore from above results we get $yz+zx+xy=\frac{3731}{900}$