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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 $a + b$.

 Jun 13, 2024
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We know that the slope of AB is 2, which means we can write a formula. 

 

The slope of a line is \(\frac{y_2-y_1}{x_2-x_1}\). Plugging in the points \((a, a^2)\) and \((b, b^2)\), we can write the equation

\(\text{Slope} = \frac{b^2 - a^2 }{b - a}\)

\(\frac{b^2-a^2}{b-a}=2\)

 

Knowing that \(b^2-a^2=(b-a)(b+a)\), we have

\(\frac{ (b -a) (b + a) }{(b -a)} = b + a = 2\)

 

So our final answer is just 2. 

 

Thanks! :)

 Jun 13, 2024

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