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 $a + b$.

learnmgcat Jul 11, 2024

#1**+1 **

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! :)

NotThatSmart Jul 11, 2024