Question

A rod of fixed length $k$ has its ends sliding along the coordinates axes with 1st quadrant such that rod meets the x-axis at $A(a,0)$ and y-axis at $B(0,b)$. If $O$ is origin and $E={(a+\frac{1}{b})}^2+{(b+\frac{1}{a})}^2$. Which of the following holds good?

• A. Minimum value of $E$ is $k^2+\frac{4}{k^2}+2$
• B. Minimum value of $E$ is ${(k+\frac{2}{k})}^2$
• C. When $E$ is minimum then area of $\Delta AOB$ is $\frac{k^2}{4}$
• D. When $E$ is minimum then area of $\Delta AOB$ is $\frac{k^2}{2}$

Answer 1

The correct option is A Minimum value of $E$ is $k^2+\frac{4}{k^2}+2$

Now, from the figure.

$E={\left(a+\frac{1}{b}\right)}^2+{\left(b+\frac{1}{a}\right)}^2$

minimum value of $E=k^2+\frac{4}{k^2}+2$

Hence, the option $\left(A\right)$ is correct.