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Difference between revisions of "1991 AHSME Problems/Problem 18"
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== Problem ==
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If <math>S</math> is the set of points <math>z</math> in the complex plane such that <math>(3+4i)z</math> is a real number, then <math>S</math> is a  
 
If <math>S</math> is the set of points <math>z</math> in the complex plane such that <math>(3+4i)z</math> is a real number, then <math>S</math> is a  
  
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(D) line
 
(D) line
 
(E) parabola
 
(E) parabola
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== Solution ==
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<math>\fbox{D}</math>
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Solution 1:
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We want <math>(3+4i)z</math> a real number, so we want the <math>4i</math> term to be canceled out. Then, we can make <math>z</math> be in the form <math>(n-\frac{4}{3}ni)</math> to make sure the imaginary terms cancel out when it's multiplied together. <math>(n-\frac{4}{3}ni)</math> is a line, so the answer is <math>\textbf{(D) } line</math>
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== See also ==
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{{AHSME box|year=1991|num-b=17|num-a=19}} 
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[[Category: Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:53, 11 October 2016

Problem

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a

(A) right triangle (B) circle (C) hyperbola (D) line (E) parabola

Solution

$\fbox{D}$

Solution 1:

We want $(3+4i)z$ a real number, so we want the $4i$ term to be canceled out. Then, we can make $z$ be in the form $(n-\frac{4}{3}ni)$ to make sure the imaginary terms cancel out when it's multiplied together. $(n-\frac{4}{3}ni)$ is a line, so the answer is $\textbf{(D) } line$

See also

1991 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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
All AHSME Problems and Solutions

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