Mathematics

Mathematics

Problem

If ∣z−2∣/∣z−3∣=2|z - 2|/|z - 3| = 2 represents a circle, then its radius is equal to a. 1 b. 13\frac{1}{3} c. 34\frac{3}{4} d. 23\frac{2}{3}
Intuitive explanation

The key is recognizing that a ratio of distances from a point to two fixed points traces out a circle—this is the Apollonius circle. Rearrange the equation by clearing the fraction and completing the square to find the center and radius. The algebra will reveal that all points zz satisfying this condition lie equidistant from a single center, which defines your circle.

Intuitive explanation

The key is recognizing that a ratio of distances from a point to two fixed points traces out a circle—this is the Apollonius circle. Rearrange the equation by clearing the fraction and completing the square to find the center and radius. The algebra will reveal that all points zz satisfying this condition lie equidistant from a single center, which defines your circle.

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