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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 rr be the distance from the origin to a point PP with coordinates xx and y.y. Designate the ratio yr\frac{y}{r} by ss and the ratio xr\frac{x}{r} by c.c. Then the values of s2−c2s^2-c^2 are limited to the numbers:

less than −1-1 and greater than +1,+1, both excluded

less than −1-1 and greater than +1,+1, both included

between −1-1 and +1,+1, both excluded

between −1-1 and +1,+1, both included

−1-1 and +1+1 only

Answer: D
Concepts:coordinate geometryPythagorean Theoreminequality
Difficulty rating: 1280
Small Hint:

Use r2=x2+y2r^2=x^2+y^2 to relate s2+c2s^2+c^2

Big Hint:

Rewrite s2−c2s^2-c^2 as 2s2−12s^2-1

Solution:

Since r2=x2+y2,r^2=x^2+y^2, s2+c2=y2+x2r2=1. s^2+c^2=\frac{y^2+x^2}{r^2}=1. Thus s2−c2=2s2−1.s^2-c^2=2s^2-1. Because 0≤s2≤1,0\le s^2\le1, this expression ranges from −1-1 through 1,1, with both endpoints included.

Thus, the correct answer is D.

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