A(z1=2+2i), B(z2), C(z3) are three points on the Argand plane satisfying |zk−2i|=2, (k=1,2,3). If ΔABC encloses the maximum area, then the sum of the imaginary parts of z2 and z3 is…….

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$A (z_{1} = 2 + 2 i), B (z_{2}), C (z_{3})$ are three points on the Argand plane satisfying $| z_{k} - 2 i | = 2$ , $(k = 1, 2, 3)$ . If $Δ A B C$ encloses the maximum area, then the sum of the imaginary parts of $z_{2}$ and $z_{3}$ is…….

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answer is 4.

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Detailed Solution

$| z_{k} - 2 i | = 2$ represents a circle with centre (0,2) and radius 2

Let $B = (x_{2}, y_{2}), C = (x_{3}, y_{3})$

Area of triangle ABC inscribed in a circle is maximum if the triangle is equilateral

Equation of $\overset{\leftrightarrow}{A D}$ is $y = 2$ , which is perpendicular bisector of BC

D = midpoint of $B C = (\frac{x_{2} + x_{3}}{2}, \frac{y_{2} + y_{3}}{2})$ , which lies on $y = 2 \Rightarrow \frac{y_{2} + y_{3}}{2} = 2 \Rightarrow y_{2} + y_{3} = 4$