2023 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra Geometry

Points

A

and

B

lie on the graph of

y=\log_2 x.

The midpoint of

\overline{AB}

is

(6,2).

What is the positive difference between the

x

-coordinates of

A

and

B?

Show solution

Solution

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,

so

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.

Tagged: logarithm · midpoint · algebraic manipulation

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