Question

If $a,b>0$ satisfy $a^3+b^3=a-b$, then

• A. $a^2+b^2>1$
• B. $a^2–b<0$
• C. $a^2+b^2=1$
• D. $a^2+ab+b^2<1$

Answer 1

The correct option is D

$a^2+ab+b^2<1$

Explanation for the correct option:

Given, $a^3+b^3=a–b$

Let $a=\frac{2}{3},b=\frac{1}{3}$

${\left(\frac{2}{3}\right)}^3+{\left(\frac{1}{3}\right)}^3=\frac{2}{3}–\frac{1}{3}$

$\Rightarrow$ $\frac{8}{27}+\frac{1}{27}=\frac{1}{3}$

$\Rightarrow$ $\frac{1}{3}=\frac{1}{3}$

$\begin{array}{rcl}\therefore a^2+ab+b^2& =& \frac{4}{9}+\frac{2}{3}\times \frac{1}{3}+\frac{1}{9}\\ & =& \frac{4}{9}+\frac{2}{9}+\frac{1}{9}\\ & =& \frac{\mathbf{7}}{\mathbf{9}}\mathbf{}\mathbf{<}\mathbf{}\mathbf{1}\end{array}$

Hence, Option ‘D’ is Correct.