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avatar+1222 

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 $a + b$.

 Jul 11, 2024
 #1
avatar+1910 
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Let's first focus on the slope of the line introduced. 

First, note that the slope of a line is in the form \(\frac{y_2-y_1}{x_2-x_1}\)

 

Thus, plugging in the two points we have, we get

\( [ b^2 - a^2 ] / [ b - a ]=2\)

 

Now, using the difference of squares thereom, we can simpplify the top to

\( [ (b -a) (b + a) ] / (b -a) = 2\)

 

The b-a cancels out, leaving us with

\( b + a = 2\)

 

So 2 is our answer. 

 

Thanks! :)

 Jul 11, 2024

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