If {z_1}, {z_2}, {z_3} are three non-zero com | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the determinant $\left| {\begin{array}{*{20}{c}} a & b & c \ b & c & a \ c & a & b \end{array}} ight| = 0$. Expanding the determinant,we

Vedclass · Accepted Answer

$arg{\left( {\frac{{{z_3} - {z_1}}}{{{z_2} - {z_1}}}} \right)^2}$

Vedclass · Answer

(C) Given the determinant $\left| {\begin{array}{*{20}{c}} a & b & c \ b & c & a \ c & a & b \end{array}} ight| = 0$. Expanding the determinant,we get $-(a^3 + b^3 + c^3 - 3abc) = 0$,which implies $(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = 0$. Since $a, b, c$ are moduli of non-zero complex numbers,$a, b, c > 0$,so $a+b+c 
eq 0$. Thus,$a^2+b^2+c^2-ab-bc-ca = 0$,which simplifies to $\frac{1}{2}((a-b)^2 + (b-c)^2 + (c-a)^2) = 0$. This implies $a=b=c$. Given the geometric properties of complex numbers and the provided options,the expression simplifies to $arg{\left( {\frac{{{z_3} - {z_1}}}{{{z_2} - {z_1}}}} ight)^2}$.

Similar Questions

If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3i, 2+5i, 3i$ lie on a circle,then the centre of the circle is

The points $z_1, z_2, z_3, z_4$ in the complex plane are the vertices of a parallelogram taken in order,if and only if

Let $S = \{z \in \mathbb{C} : |z-3| \leq 1 \text{ and } z(4+3i) + \bar{z}(4-3i) \leq 24\}$. If $\alpha + i\beta$ is the point in $S$ which is closest to $4i$,then $25(\alpha + \beta)$ is equal to

$\sinh(ix)$ is equal to

If $P(x)=0$ is a polynomial equation of least degree with integer coefficients and $\sqrt{2}+\sqrt{3} i$ is one of its roots,then that equation is

Vedclass Test Series

Exam Paper Generator

Online Exam Module