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If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3 i, 2+5 i, 3 i$ lie on a circle, then the centre of the circle is
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The correct answer is:
$2+3 i$
Four points $A, B, C, D$ on a Circle are
$(2,1),(4,3),(2,5),(0,3)$
Slope of $A B=\frac{3-1}{4-2}=\frac{2}{2}=1$
Slope of $B C=\frac{5-3}{2-4}=-1$
$A B C$ is a right angle $\Delta$
$\therefore$ Centre of circle is mid-point of $A C$ i.e., $\left(\frac{2+2}{2}, \frac{1+5}{2}\right)=(2,3)$ $\therefore$ Centre of circle is $(2+3 i)$
$(2,1),(4,3),(2,5),(0,3)$
Slope of $A B=\frac{3-1}{4-2}=\frac{2}{2}=1$
Slope of $B C=\frac{5-3}{2-4}=-1$
$A B C$ is a right angle $\Delta$
$\therefore$ Centre of circle is mid-point of $A C$ i.e., $\left(\frac{2+2}{2}, \frac{1+5}{2}\right)=(2,3)$ $\therefore$ Centre of circle is $(2+3 i)$
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