+228 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 $a + b$.
+1826 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! :)
