2010 AMC 12A Problem 13

Attempt Problem 13 of the 2010 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12A solutions, or check the answer key.

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13.

For how many integer values of

k

do the graphs of

x^2+y^2=k^2

and

xy=k

not intersect?

0

1

2

4

8

For

k\ne0,

x^2+y^2=k^2

is a circle of radius

|k|

and

xy=k

is a hyperbola

The hyperbola’s points closest to the origin are at distance

\sqrt{2|k|};

compare with the radius

|k|

For

k=0,

the graph of

x^2+y^2=0

is the single point

(0,0)

and

xy=0

is the two axes, which meet at the origin, so the graphs intersect.

For

k\ne0,

the circle has radius

|k|,

and the hyperbola

xy=k

has its two vertices nearest the origin at distance

\sqrt{2|k|}.

The graphs meet exactly when

|k|\ge\sqrt{2|k|},

that is

|k|\ge2.

So they fail to intersect only when

|k|=1,

namely

k=1

and

k=-1,

giving

2

values.

Thus, C is the correct answer.

Problem 13 in other years