+331 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 $a + b$.
+1950 Alright, not too bad of a problem!
So, we just have to set up the slope equation to find what a+b is equal to because spoiler alert, this is a really easy problem once we write out the equation.
The slope formula is where the slope of a line is equal to x2−x1y2−y1 where (x2, y2) and (x1, y1) are points on that line.
In this case, we have point 1 as (a,a^2) and point 2 as (b,b^2).
Plugging these numbers into the formula, we have b2−a2b−a=2. Now, there's on trick that helps us solve thus problem immediately. That is, Mr. Good Old Difference of Squares.
We can write the numerator as (b−a)(b+a), meaning we can just cancel out the b - a.
(b−a)(b+a)b−a=b+a=2
Boom, this means b + a = 2, so our final answer is simply 2! Simple enough!
Thanks!
