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2004 AMC 12B Problem 16

Problem 16 of 25IntermediateAlgebraGeometry

A function ff is defined by f(z)=iz‾,f(z) = i\overline{z}, where i=−1i = \sqrt{-1} and z‾\overline{z} is the complex conjugate of z.z. How many values of zz satisfy both ∣z∣=5|z| = 5 and f(z)=z?f(z) = z?

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Solution

Writing z=x+iy,z = x + iy, we get f(z)=i(x−iy)=y+ix.f(z) = i(x - iy) = y + ix. Setting f(z)=zf(z) = z gives y=x,y = x, which is a line through the origin. The condition ∣z∣=5|z| = 5 is a circle, and a line through the center meets the circle in 22 points. Thus, the correct answer is C.

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Concepts: complex number · coordinate geometry

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