Points A and B lie on the graph of y=log_2 x. The midpoint of AB is (6,2). What is the positive difference between the x -coordinates of A and B?

2023 AMC 12A Problem 6

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Points

A

and

B

lie on the graph of

y=\log_2 x.

The midpoint of

\overline{AB}

(6,2).

What is the positive difference between the

x

-coordinates of

A

and

B?

2\sqrt{11}

4\sqrt{3}

8

4\sqrt{5}

9

The

x

-coordinates sum to

2\cdot 6=12

The

y

-coordinates average to

2,

\log_2 x_1+\log_2 x_2=4,

giving

x_1x_2=16

Let the

x

-coordinates be

x_1

and

x_2.

The midpoint gives

x_1+x_2=12,

and the average of the

y

-values gives

\log_2 x_1+\log_2 x_2=4,

x_1x_2=2^4=16.

Then

\begin{gathered} |x_1-x_2|\\ {}=\sqrt{(x_1+x_2)^2-4x_1x_2}\\ {}=\sqrt{144-64}\\ {}=\sqrt{80}\\ {}=4\sqrt5. \end{gathered}

Thus, the correct answer is D.