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> | ||
− | + | == 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>. | |
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− | Suppose we pick an arbitrary point on the [[complex plane]], say <math>(1,1)</math>. According to the definition of <math>f(z) | ||
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== |
+ | 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 == | |
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> | ||
\begin{eqnarray*} |(a + bi)z - z| & = & |(a + bi)z| \\ | \begin{eqnarray*} |(a + bi)z - z| & = & |(a + bi)z| \\ | ||
− | |z(a - 1) | + | |z(a - 1 + bi)| & = & |z(a + bi)| \\ |
− | |z||(a - 1) + bi| & = & |z||a + bi| \\ | + | |z|\cdot|(a - 1) + bi| & = & |z|\cdot|a + bi| \\ |
+ | |(a - 1) + bi| & = & |a + bi| \\ | ||
(a - 1)^2 + b^2 & = & a^2 + b^2 \\ | (a - 1)^2 + b^2 & = & a^2 + b^2 \\ | ||
& \Rightarrow & a = \frac 12 \end{eqnarray*} | & \Rightarrow & a = \frac 12 \end{eqnarray*} | ||
</cmath> | </cmath> | ||
− | Since <math>|a + bi| = 8,</math> <math>a^2 + b^2 = 64.</math> | + | Since <math>|a + bi| = 8,</math> <math>a^2 + b^2 = 64.</math> Because <math>a = \frac 12,</math> thus <math>b^2 = \frac {255}4.</math> So the answer is <math>\boxed{259}</math>. |
− | + | == 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>. | ||
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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>. | ||
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+ | ~ anellipticcurveoverq | ||
== See also == | == See also == |
Latest revision as of 10:58, 8 June 2021
Problem
A function is defined on the complex numbers by where and 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 and that where and are relatively prime positive integers, find
Solution 1
Suppose we pick an arbitrary point on the complex plane, say . According to the definition of , this image must be equidistant to and . Thus the image must lie on the line with slope and which passes through , so its graph is . Substituting and , we get .
By the Pythagorean Theorem, we have , and the answer is .
Solution 2
Plugging in yields . This implies that must fall on the line , given the equidistant rule. By , we get , and plugging in yields . The answer is thus .
Solution 3
We are given that is equidistant from the origin and This translates to Since Because thus So the answer is .
Solution 4
Let and be the points in the complex plane represented by and , respectively. implies . Also, we are given , so is isosceles with base . Notice that the base angle of this isosceles triangle is equal to the argument of the complex number , because forms an angle of with . Drop the altitude/median from to base , and you end up with a right triangle that shows . Since and are positive, lies in the first quadrant and ; hence by right triangle trigonometry . Finally, , and , so the answer is .
Solution 5
Similarly to in Solution 3, we see that . Letting the point , we have . Expanding both sides of this equation (after squaring, of course) and canceling terms, we get . Of course, can't be zero because this property of the function holds for all complex . Therefore, and we proceed as above to get .
~ anellipticcurveoverq
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 | ||
All AIME Problems and Solutions |
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