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# Geometry

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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 $a + b$.

May 3, 2024

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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 $$\frac{x2-x1}{y2 - y1}$$ 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 $$\frac{ b^2 - a^2}{b - a} = 2$$. 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.

$$\frac{(b -a) (b + a)}{b -a} = b + a = 2$$

Boom, this means b + a = 2, so our final answer is simply 2! Simple enough!

Thanks!

May 3, 2024