CAT 2019 Slot 1 QA Question & Solution
AlgebraMedium
Question
Let x and y be positive real numbers such that
$\log_{5}{(x + y)} + \log_{5}{(x - y)} = 3,$ and $\log_{2}{y} - \log_{2}{x} = 1 - \log_{2}{3}$. Then $xy$ equals
Options
150
25
100
250
Solution
We have, $\log_{5}{(x + y)} + \log_{5}{(x - y)} = 3$
=> $x^2-y^2=125$......(1)
$\log_{2}{y} - \log_{2}{x} = 1 - \log_{2}{3}$
=>$\ \frac{\ y}{x}$ = $\ \frac{\ 2}{3}$
=> 2x=3y => x=$\ \frac{\ 3y}{2}$
On substituting the value of x in 1, we get
$\ \frac{\ 5x^2}{4}$=125
=>y=10, x=15
Hence xy=150