+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 $a + b$.
+1908 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! :)
