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Let A = 2 + 3i and B = 8 + 11i. A complex number z varies so that the…
Question
Let A = 2 + 3i and B = 8 + 11i. A complex number z varies so that the argument of (z - A)/(z - B) is constantly equal to pi/3. This locus is an arc of a circle of radius R. Compute the value of 3*R^2. Report the exact integer.
✓ Verified answer: 100checked by our engine — not a guess
Step-by-step solution
The condition arg((z - A)/(z - B)) = pi/3 means the chord AB subtends a constant inscribed angle pi/3 at the moving point z, so z lies on a circular arc.
By the inscribed-angle / extended law of sines, the chord length and the inscribed angle relate by |AB| = 2R sin(angle).
Here |B - A| = |6 + 8i| = sqrt(36 + 64) = 10.
With the inscribed angle pi/3, sin(pi/3) = sqrt(3)/2, so 10 = 2R*(sqrt3/2) = R*sqrt3, giving R = 10/sqrt3 and R^2 = 100/3.
Therefore 3R^2 = 100.
Final answer100
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