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

 
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A function <math>f</math> is defined on the [[complex number]]s by <math>f(z)=(a+bi)z,</math> where <math>a_{}</math> and <math>b_{}</math> 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 <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers,  find <math>m+n.</math>
 
A function <math>f</math> is defined on the [[complex number]]s by <math>f(z)=(a+bi)z,</math> where <math>a_{}</math> and <math>b_{}</math> 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 <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers,  find <math>m+n.</math>
  
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== Solution 1 ==
 
 
=== Solution 1 ===
 
 
Suppose we pick an arbitrary point on the [[complex plane]], say <math>(1,1)</math>. According to the definition of <math>f(z)</math>,  <cmath>f(1+i) = (a+bi)(1+i) = (a-b) + (a+b)i,</cmath> this image must be equidistant to <math>(1,1)</math> and <math>(0,0)</math>. Thus the image must lie on the line with slope <math>-1</math> and which passes through <math>\left(\frac 12, \frac12\right)</math>, so its graph is <math>x + y = 1</math>. Substituting <math>x = (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.  
 
Suppose we pick an arbitrary point on the [[complex plane]], say <math>(1,1)</math>. According to the definition of <math>f(z)</math>,  <cmath>f(1+i) = (a+bi)(1+i) = (a-b) + (a+b)i,</cmath> this image must be equidistant to <math>(1,1)</math> and <math>(0,0)</math>. Thus the image must lie on the line with slope <math>-1</math> and which passes through <math>\left(\frac 12, \frac12\right)</math>, so its graph is <math>x + y = 1</math>. Substituting <math>x = (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.  
  
 
By the [[Pythagorean Theorem]], we have <math>\left(\frac{1}{2}\right)^2 + b^2 = 8^2 \Longrightarrow b^2 = \frac{255}{4}</math>, and the answer is <math>\boxed{259}</math>.
 
By the [[Pythagorean Theorem]], we have <math>\left(\frac{1}{2}\right)^2 + b^2 = 8^2 \Longrightarrow b^2 = \frac{255}{4}</math>, and the answer is <math>\boxed{259}</math>.
  
===Solution 2===
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==Solution 2==
 
Plugging in <math>z=1</math> yields <math>f(1) = a+bi</math>. This implies that <math>a+bi</math> must fall on the line <math>Re(z)=a=\frac{1}{2}</math>, given the equidistant rule. By <math>|a+bi|=8</math>, we get <math>a^2 + b^2 = 64</math>, and plugging in <math>a=\frac{1}{2}</math> yields <math>b^2=\frac{255}{4}</math>. The answer is thus <math>\boxed{259}</math>.
 
Plugging in <math>z=1</math> yields <math>f(1) = a+bi</math>. This implies that <math>a+bi</math> must fall on the line <math>Re(z)=a=\frac{1}{2}</math>, given the equidistant rule. By <math>|a+bi|=8</math>, we get <math>a^2 + b^2 = 64</math>, and plugging in <math>a=\frac{1}{2}</math> yields <math>b^2=\frac{255}{4}</math>. The answer is thus <math>\boxed{259}</math>.
  
=== Solution 3 ===
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== Solution 3 ==
 
We are given that <math>(a + bi)z</math> is equidistant from the origin and <math>z.</math>  This translates to
 
We are given that <math>(a + bi)z</math> is equidistant from the origin and <math>z.</math>  This translates to
 
<cmath>
 
<cmath>
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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>.
 
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 4 ===
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== Solution 4 ==
 
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>.
 
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>.
  
=== Solution 5 ===  
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== Solution 5 ==
 
Similarly to in Solution 3, we see that <math>|(a + bi)z - z| = |(a + bi)z|</math>. Letting the point <math>z = c + di</math>, we have <math>\sqrt{(ab+bc-d)^2+(ac-bd-c)^2} = \sqrt{(ac-bd)^2+(ad+bc)^2}</math>. Expanding both sides of this equation (after squaring, of course) and canceling terms, we get <math>(d^2+c^2)(-2a+1) = 0</math>. Of course, <math>(d^2+c^2)</math> can't be zero because this property of the function holds for all complex <math>z</math>. Therefore, <math>a = \frac{1}{2}</math> and we proceed as above to get <math>\boxed{259}</math>.
 
Similarly to in Solution 3, we see that <math>|(a + bi)z - z| = |(a + bi)z|</math>. Letting the point <math>z = c + di</math>, we have <math>\sqrt{(ab+bc-d)^2+(ac-bd-c)^2} = \sqrt{(ac-bd)^2+(ad+bc)^2}</math>. Expanding both sides of this equation (after squaring, of course) and canceling terms, we get <math>(d^2+c^2)(-2a+1) = 0</math>. Of course, <math>(d^2+c^2)</math> can't be zero because this property of the function holds for all complex <math>z</math>. Therefore, <math>a = \frac{1}{2}</math> and we proceed as above to get <math>\boxed{259}</math>.
  

Latest revision as of 20:24, 17 April 2021

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 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

Plugging in $z=1$ yields $f(1) = a+bi$. This implies that $a+bi$ must fall on the line $Re(z)=a=\frac{1}{2}$, given the equidistant rule. By $|a+bi|=8$, we get $a^2 + b^2 = 64$, and plugging in $a=\frac{1}{2}$ yields $b^2=\frac{255}{4}$. The answer is thus $\boxed{259}$.

Solution 3

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 4

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 $259$.

Solution 5

Similarly to in Solution 3, we see that $|(a + bi)z - z| = |(a + bi)z|$. Letting the point $z = c + di$, we have $\sqrt{(ab+bc-d)^2+(ac-bd-c)^2} = \sqrt{(ac-bd)^2+(ad+bc)^2}$. Expanding both sides of this equation (after squaring, of course) and canceling terms, we get $(d^2+c^2)(-2a+1) = 0$. Of course, $(d^2+c^2)$ can't be zero because this property of the function holds for all complex $z$. Therefore, $a = \frac{1}{2}$ and we proceed as above to get $\boxed{259}$.

~ anellipticcurveoverq

See also

1999 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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