Let a,b 0 satisfy a^(3)+b^(3)=a-b, then (a) a^(2)+b^(2)=1 (b) a^(2)+ab+b^(2) 1 (d) none of these

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Let  a,b > 0 satisfy  a^(3)+b^(3)=a-b, then (a)  a^(2)+b^(2)=1 (b)  a^(2)+ab+b^(2) < 1 (c)  a^(2)+b^(2) > 1 (d) none of these

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Calculate using identity: Calculate $a^3 + b^3$ using the identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. - Given: $a^3 + b^3 = a - b$ - Identity: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ - Set equal: $(a + b)(a^2 - ab + b^2) = a - b$Simplify the equation: Simplify the equation assuming $a + b \neq 0$ (since a, b > 0). - Divide both sides by $a + b$: $a^2 - ab + b^2 = \frac{a - b}{a + b}$Check option (a): Check each option by substituting and simplifying. (a) $a^2 + b^2 = 1$ - $a^2 + b^2 = 1$ implies $a^2 - ab + b^2 = 1 - ab$ - Compare: $1 - ab = \frac{a - b}{a + b}$ - This equation is not generally true without specific values for a and b.Check option (b): Check option (b). (b) $a^2 + ab + b^2 < 1$ - $a^2 + ab + b^2 = (a^2 - ab + b^2) + 2ab$ - Substitute: $\frac{a - b}{a + b} + 2ab < 1$ - This inequality depends on the values of a and b, not generally provable without additional information.Check option (c): Check option (c). (c) $a^2 + b^2 > 1$ - $a^2 + b^2 = (a^2 - ab + b^2) + ab$ - Substitute: $\frac{a - b}{a + b} + ab > 1$ - This inequality also depends on specific values of a and b, not generally provable.Conclude the results: Conclude that none of the options can be universally proven with the given information. - None of the options (a), (b), or (c) can be definitively proven true for all positive a and b satisfying the original equation.

$1$ . Let a, b>0 satisfy $a^{3}+b^{3}=a-b$ , then $\newline$ (a) $a^{2}+b^{2}=1$ $\newline$ (b) a^{2}+a b+b^{2}<1 $\newline$ (c) a^{2}+b^{2}>1 $\newline$ (d) none of these

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111. Let a, b>0 satisfy a3+b3=a−b a^{3}+b^{3}=a-b a3+b3=a−b, then\newline(a) a2+b2=1 a^{2}+b^{2}=1 a2+b2=1\newline(b) a^{2}+a b+b^{2}<1 \newline(c) a^{2}+b^{2}>1 \newline(d) none of these

$1$ . Let a, b>0 satisfy $a^{3}+b^{3}=a-b$ , then $\newline$ (a) $a^{2}+b^{2}=1$ $\newline$ (b) a^{2}+a b+b^{2}<1 $\newline$ (c) a^{2}+b^{2}>1 $\newline$ (d) none of these