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Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$.  The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$.  Find $a + b$.

 Apr 30, 2024
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Using the slope formula gives \(\dfrac{b^2 - a^2}{b - a} = 2\).

 

Note that \(b^2 - a^2 = (b - a)(a + b)\) by difference of squares formula. Then, simplifying, we have

 

\(\dfrac{(b - a)(a + b)}{b - a} = 2\\ a + b = \boxed{2}\)

 Apr 30, 2024

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