Let z=x+yi,where x, y are integers and i=sqrt | vedclass

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

Question

(C) Given the equation $\bar{z}z^3+z(\bar{z})^3=700$. Since $z=x+iy$,we have $\bar{z}=x-iy$ and $z\bar{z}=x^2+y^2$. The equation can be written as $\b

Vedclass · Accepted Answer

$48$

Vedclass · Answer

(C) Given the equation $\bar{z}z^3+z(\bar{z})^3=700$. Since $z=x+iy$,we have $\bar{z}=x-iy$ and $z\bar{z}=x^2+y^2$. The equation can be written as $\bar{z}z(z^2+(\bar{z})^2)=700$. Substituting $z=x+iy$ and $\bar{z}=x-iy$,we get $(x^2+y^2)((x+iy)^2+(x-iy)^2)=700$. $(x^2+y^2)(x^2-y^2+2ixy+x^2-y^2-2ixy)=700$. $(x^2+y^2)(2(x^2-y^2))=700$. $(x^2+y^2)(x^2-y^2)=350$. Since $x, y$ are integers,we look for factors of $350$. We have $x^2+y^2=25$ and $x^2-y^2=14$ (not integer solutions) or $x^2+y^2=50$ and $x^2-y^2=7$ (not integer solutions). Wait,let us re-evaluate: $x^2+y^2=25$ and $x^2-y^2=7$ gives $2x^2=32$ $\Rightarrow x^2=16$ $\Rightarrow x=\pm 4$ and $2y^2=18$ $\Rightarrow y^2=9$ $\Rightarrow y=\pm 3$. The vertices are $(\pm 4, \pm 3)$. The rectangle has length $2|x| = 8$ and width $2|y| = 6$. Area $= 8 	imes 6 = 48$. Thus,option $C$ is correct.

Let $z=x+yi$,where $x, y$ are integers and $i=\sqrt{-1}$. The area of the rectangle whose vertices are the roots of the equation $\bar{z}z^3+z(\bar{z})^3=700$ is

Similar Questions

Let $z=x+iy$ and $P(x, y)$ be a point on the Argand plane. If $z$ satisfies the condition $\operatorname{Arg}\left(\frac{z-3i}{z+2i}\right)=\frac{\pi}{4}$, then the locus of $P$ is:

If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

If $z = x + iy$ represents a point in the Argand plane,then a point which is not in the region represented by $|z - 1 + i| \leq 2$ is

$S = \{z \in \mathbb{C} : |z - 1 + i| = 1\}$ represents

The roots of the cubic equation $(z + ab)^3 = a^3$,where $a \neq 0$,represent the vertices of a triangle of sides of length:

Vedclass Test Series

Exam Paper Generator

Online Exam Module