Question

Find the general values of $x$ and $y$ which make following expression a perfect square $5x^2+y^2$.

• A. $x=4mn,y=4m^2-n^2$
• B. $x=3mn,y=5m^2-2n^2$
• C. $x=2mn,y=5m^2-n^2$
• D. None of these

Answer 1

Answer: (C)

$x=2mn,y=5m^2-n^2$

$f(x,y)=5x^2+y^2$

We know that ${(a-b)}^2+4ab={(a+b)}^2$

Let $y=a-b,x=2\sqrt{\frac{ab}{5}}$

$\therefore f(x,y)=5x^2+y^2=\left(5\right)\left(4\left(\frac{ab}{5}\right)\right)+{(a-b)}^2$

$={(a-b)}^2+4ab$

$\therefore$Let $a=5m^2,b=n^2$

$\therefore y=5m^2-n^2,x=2mn$ is a general solution of $f(x,y)$