The points in the argand plane represented by the complex conjugates of \(1+2 i, 2-3 i, 3-4 i\)
The points in the argand plane represented by the complex conjugates of \(1+2 i, 2-3 i, 3-4 i\)
are collinear
form an equilateral triangle
form an obtuse angled triangle
form an acute angled triangle
Solution
The complex conjugate of given points are in argand plane \(A(1-2 i), B(2+3 i), C(3+4 i)\)
So
\(\begin{aligned}
a & =B C=\sqrt{1+1}=\sqrt{2} \\
b & =C A=\sqrt{4+36}=\sqrt{40} \\
c & =A B=\sqrt{1+25}=\sqrt{26} \\
\because \quad \cos B & =\frac{a^2+c^2-b^2}{2 a c}=\frac{2+26-40}{2 \sqrt{2} \sqrt{26}} \\
& =-\frac{12}{4 \sqrt{13}}=-\frac{3}{\sqrt{13}} < 0
\end{aligned}\)
So, point \(A, B\) and \(C\) represents the vertices of an obtuse angled triangle.
Hence, option (c) is correct.