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 $a + b$.

ABJeIIy Jul 21, 2024

#1**+1 **

First, we know the equation of a sliope is \(\frac{y_2-y_1}{x_2-x_1}\)

In this problem, we know the two points we have are (a, a^2) and (b, b^2).

Plugging these two points in, we get

\( [ b^2 - a^2 ] / [ b - a ]\)

Using the difference of squares equation for the numerator, we have

\( [ (b -a) (b + a) ] / (b -a)\)

Notice the b - a cancel each other out.

This leaves us with

\( b + a \)

Since the slope is 2, we simplfy have

\( b + a = 2\)

So the answer is 2.

Thanks! :)

NotThatSmart Jul 21, 2024

#1**+1 **

Best Answer

First, we know the equation of a sliope is \(\frac{y_2-y_1}{x_2-x_1}\)

In this problem, we know the two points we have are (a, a^2) and (b, b^2).

Plugging these two points in, we get

\( [ b^2 - a^2 ] / [ b - a ]\)

Using the difference of squares equation for the numerator, we have

\( [ (b -a) (b + a) ] / (b -a)\)

Notice the b - a cancel each other out.

This leaves us with

\( b + a \)

Since the slope is 2, we simplfy have

\( b + a = 2\)

So the answer is 2.

Thanks! :)

NotThatSmart Jul 21, 2024