1958 AMC 12 Problem 38

Attempt Problem 38 of the 1958 AMC 12 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 1958 AMC 12 solutions, or check the answer key.

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38.

Let

r

be the distance from the origin to a point

P

with coordinates

x

and

y.

Designate the ratio

\frac{y}{r}

by

s

and the ratio

\frac{x}{r}

by

c.

Then the values of

s^2-c^2

are limited to the numbers:

less than

-1

and greater than

+1,

both excluded

less than

-1

and greater than

+1,

both included

between

-1

and

+1,

both excluded

between

-1

and

+1,

both included

-1

and

+1

only

Answer: D

Concepts:coordinate geometry Pythagorean Theorem inequality

Difficulty rating: 1280

Small hint:

Use

r^2=x^2+y^2

to relate

s^2+c^2

Big hint:

Rewrite

s^2-c^2

as

2s^2-1

Solution:

Since

r^2=x^2+y^2,

s^2+c^2=\frac{y^2+x^2}{r^2}=1.

Thus

s^2-c^2=2s^2-1.

Because

0\le s^2\le1,

this expression ranges from

-1

through

1,

with both endpoints included.

Thus, the correct answer is D.

Problem 38 in other years