Question

If $x,y$ are two positive numbers such that $x-y=10$ and square of the greater exceeds twice the square of the smaller by maximum value, then which of the following is/are true

• A. $x^2-y^2=200$
• B. $xy=200$
• C. $x^3+y^3=900$
• D. $x^2+y^2=500$

Answer 1

Answer: (B), (D)

$x^2+y^2=500$

Given, $x-y=10$
Let $f=x^2-2y^2$ (to be maximum)
$\Rightarrow f\left(y\right)=(10+y{)}^2-2y^2=-y^2+20y+100$
$f^{\prime }\left(y\right)=-2y+20$ and $f^{\prime \prime }\left(y\right)=-2<0$
$\therefore f\left(y\right)$ is maximum when $f^{\prime }\left(y\right)=0$
$\Rightarrow y=10$
$\Rightarrow x=20$
So, $x^2+y^2=500,x^2-y^2=300$
$x^3+y^3=9000,xy=200$