Question

Let ordered pair ($\alpha$, $\beta$) satisfying the system of equations
2 log ($x^2$ + $y^2$) log 5 = log {2($x^2$ + $y^2$) + 75} and log$\left(\frac{x}{3}\right)$ + log(5y) = 1 + log 2 , then

• A. Number of such ordered pair is 2
• B. ${\alpha }^2$ + ${\beta }^2$ = 25
• C. $\alpha$ and $\beta$ are prime numbers
• D. $\alpha$ and $\beta$ are coprime numbers

Answer 1

The correct option is A Number of such ordered pair is 2

Given,

$log\left(\frac{x}{3}\right)+log\left(5y\right)=1+log2$

$log\left(\frac{5xy}{3}\right)=log2$

$⟹xy=12$ and Domain $x>0$

$y>0$

Also, $2log(x^2+y^2)-log5=log\left[2\right(x^2+y^2)+75]$

Let $x^2+y^2=t$

$⟹log{t}^2-log5=log(2t+75$

$⟹\frac{t^2}{5}=2t+75$

$⟹t^2-10t-375=0$

$⟹t=25,-15$

$\therefore x^2+y^2=25$ and $xy=12$

On solving we get $x=3,y=4$

Hence the number of ordered pair is $2$.