Difference between revisions of "1999 AIME Problems/Problem 9"

Revision as of 23:48, 7 March 2009

Problem

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n.$

Solution

Solution 1

Suppose we pick an arbitrary point on the complex plane, say $(1,1)$ . According to the definition of $f(z) = f(1+i) = (a+bi)(1+i) = (a-b) + (a+b)i$ , this image must be equidistant to $(1,1)$ and $(0,0)$ . Thus the image must lie on the line with slope $-1$ and which passes through $\left(\frac 12, \frac12\right)$ , so its graph is $x + y = 1$ . Substituting $x = (a-b)$ and $y = (a+b)$ , we get $2a = 1 \Rightarrow a = \frac 12$ .

By the Pythagorean Theorem, we have $\left(\frac{1}{2}\right)^2 + b^2 = 8^2 \Longrightarrow b^2 = \frac{255}{4}$ , and the answer is $\boxed{259}$ .

Solution 2

We are given that $(a + bi)z$ is equidistant from the origin and $z.$ This translates to $\begin{eqnarray*} |(a + bi)z - z| & = & |(a + bi)z| \\ |z(a - 1) + bzi| & = & |az + bzi| \\ |z||(a - 1) + bi| & = & |z||a + bi| \\ (a - 1)^2 + b^2 & = & a^2 + b^2 \\ & \Rightarrow & a = \frac 12 \end{eqnarray*}$ Since $|a + bi| = 8,$ $a^2 + b^2 = 64.$ But $a = \frac 12,$ thus $b^2 = \frac {255}4.$ So the answer is $259$ .

Solution 3

Let $P$ and $Q$ be the points in the complex plane represented by $z$ and $(a+bi)z$ , respectively. $|a+bi| = 8$ implies $OQ = 8OP$ . Also, we are given $OQ = PQ$ , so $OPQ$ is isosceles with base $OP$ . Notice that the base angle of this isosceles triangle is equal to the argument $\theta$ of the complex number $a + bi$ , because $(a+bi)z$ forms an angle of $\theta$ with $z$ . Drop the altitude/median from $Q$ to base $OP$ , and you end up with a right triangle that shows $\cos \theta = \frac{\frac{1}{2}OP}{8OQ} = \frac{\frac{1}{2}|z|}{8|z|} = \frac{1}{16}$ . Since $a$ and $b$ are positive, $z$ lies in the first quadrant and $\theta < \pi/2$ ; hence by right triangle trigonometry $\sin \theta = \frac{\sqrt{255}}{16}$ . Finally, $b = |a+bi|\sin\theta = 8\frac{\sqrt{255}}{16} = \frac{\sqrt{255}}{2}$ , and $b^2 = \frac{255}{4}$ , so the answer is $259$ .

@@ Line 20: / Line 20: @@
 </cmath>
 Since <math>|a + bi| = 8,</math> <math>a^2 + b^2 = 64.</math>  But <math>a = \frac 12,</math> thus <math>b^2 = \frac {255}4.</math>  So the answer is <math>259</math>.
+=== Solution 3 ===
+Let <math>P</math> and <math>Q</math> be the points in the complex plane represented by <math>z</math> and <math>(a+bi)z</math>, respectively.  <math>|a+bi| = 8</math> implies <math>OQ = 8OP</math>.  Also, we are given <math>OQ = PQ</math>, so <math>OPQ</math> is isosceles with base <math>OP</math>.  Notice that the base angle of this isosceles triangle is equal to the argument <math>\theta</math> of the complex number <math>a + bi</math>, because <math>(a+bi)z</math> forms an angle of <math>\theta</math> with <math>z</math>.  Drop the altitude/median from <math>Q</math> to base <math>OP</math>, and you end up with a right triangle that shows <math>\cos \theta = \frac{\frac{1}{2}OP}{8OQ} = \frac{\frac{1}{2}|z|}{8|z|} = \frac{1}{16}</math>.  Since <math>a</math> and <math>b</math> are positive, <math>z</math> lies in the first quadrant and <math>\theta < \pi/2</math>; hence by right triangle trigonometry <math>\sin \theta = \frac{\sqrt{255}}{16}</math>.  Finally, <math>b = |a+bi|\sin\theta = 8\frac{\sqrt{255}}{16} = \frac{\sqrt{255}}{2}</math>, and <math>b^2 = \frac{255}{4}</math>, so the answer is <math>259</math>.
 == See also ==

1999 AIME (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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