+0

# Geometry

0
2
1
+1654

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 $a + b$.

Jul 21, 2024

#1
+1657
+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

$$b + a = 2$$

Thanks! :)

Jul 21, 2024
edited by NotThatSmart  Jul 21, 2024

#1
+1657
+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

$$b + a = 2$$