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1969 AHSME Problems/Problem 31

Revision as of 01:26, 1 October 2014 by Timneh (talk | contribs) (→‎Solution)

Problem

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$, and $v=xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:

[asy] draw((-3,0)--(3,0),EndArrow); draw((0,-4)--(0,4),EndArrow); draw((-1,0)--(0,2)--(1,0)--(0,-2)--cycle,dot); MP("(A)",(-5,2),SW); MP("O",(0,0),SW); MP("(-1,0)",(-1,0),SW); MP("(0,2)",(0,2),NE); MP("(1,0)",(1,0),SE); MP("(0,-2)",(0,-2),SE); [/asy]

[asy] draw((-3,0)--(3,0),EndArrow); draw((0,-4)--(0,4),EndArrow); draw(arc((1.5,0),2.5,126,234),black); draw(arc((-1.5,0),2.5,54,-54),black); MP("(B)",(-5,2),SW); MP("O",(0,0),SW); MP("(-1,0)",(-1,0),SW); MP("(0,2)",(0,2),NE); MP("(1,0)",(1,0),SE); MP("(0,-2)",(0,-2),SE); [/asy]


[asy] draw((-3,0)--(3,0),EndArrow); draw((0,-4)--(0,4),EndArrow); draw((-1,0)--(0,2)--(1,0),black); MP("(C)",(-5,2),SW); MP("O",(0,0),SW); MP("(-1,0)",(-1,0),SW); MP("(0,2)",(0,2),NE); MP("(1,0)",(1,0),SE); [/asy]


[asy] draw((-3,0)--(3,0),EndArrow); draw((0,-4)--(0,4),EndArrow); draw(arc((1.5,0),2.5,126,180),black); draw(arc((-1.5,0),2.5,54,0),black); MP("(D)",(-5,2),SW); MP("O",(0,0),SW); MP("(-1,0)",(-1,0),SW); MP("(0,2)",(0,2),NE); MP("(1,0)",(1,0),SE); MP("(0,-2)",(0,-2),SE); [/asy]


[asy] draw((-3,0)--(3,0),EndArrow); draw((0,-4)--(0,4),EndArrow); draw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,dot); MP("(E)",(-5,2),SW); MP("O",(.1,.1),SW); MP("(-1,0)",(-1,0),SW); MP("(0,1)",(0,1),NE); MP("(1,0)",(1,0),SE); MP("(0,-1)",(0,-1),SE); [/asy]

Solution

$\fbox{D}$

See also

1969 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 30 		Followed by
Problem 32
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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