Let the minimum value v_{0} of v = |z|^{2} + | vedclass

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

Question

(A) The function is $v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}$.Let $z = x + iy$. Then $v = (x^{2} + y^{2}) + ((x-3)^{2} + y^{2}) + (x^{2} + (y-6)^{2})$.$v

Vedclass · Accepted Answer

$1000$

Vedclass · Answer

(A) The function is $v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}$.Let $z = x + iy$. Then $v = (x^{2} + y^{2}) + ((x-3)^{2} + y^{2}) + (x^{2} + (y-6)^{2})$.$v = 3x^{2} - 6x + 9 + 3y^{2} - 12y + 36 = 3(x^{2} - 2x + 1) + 3(y^{2} - 4y + 4) + 30 = 3(x-1)^{2} + 3(y-2)^{2} + 30$.The minimum value $v_{0} = 30$ is attained at $z_{0} = 1 + 2i$.We need to calculate $|2z_{0}^{2} - \bar{z}_{0}^{3} + 3|^{2} + v_{0}^{2}$.$z_{0} = 1 + 2i$,so $z_{0}^{2} = (1+2i)^{2} = 1 - 4 + 4i = -3 + 4i$.$\bar{z}_{0} = 1 - 2i$,so $\bar{z}_{0}^{3} = (1-2i)^{3} = 1 - 3(2i) + 3(2i)^{2} - (2i)^{3} = 1 - 6i - 12 + 8i = -11 + 2i$.$2z_{0}^{2} - \bar{z}_{0}^{3} + 3 = 2(-3 + 4i) - (-11 + 2i) + 3 = -6 + 8i + 11 - 2i + 3 = 8 + 6i$.$|8 + 6i|^{2} = 8^{2} + 6^{2} = 64 + 36 = 100$.Thus,$|2z_{0}^{2} - \bar{z}_{0}^{3} + 3|^{2} + v_{0}^{2} = 100 + 30^{2} = 100 + 900 = 1000$.

Let the minimum value $v_{0}$ of $v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}$,where $z \in \mathbb{C}$,be attained at $z = z_{0}$. Then $|2z_{0}^{2} - \bar{z}_{0}^{3} + 3|^{2} + v_{0}^{2}$ is equal to:

Similar Questions

The locus of a point $z$ satisfying $|z|^2 = \operatorname{Re}(z)$ is a circle with centre

If a point $P$ denotes a complex number $z=x+iy$ in the Argand plane and if $\frac{z+1}{z+i}$ is a purely real number,then the locus of $P$ is

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$,if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$,then $|\alpha - \beta|$ is equal to $.............$.

If the amplitude of $z-2-3i$ is $\pi/4$,then the locus of $z=x+iy$ is

If $n$ is a positive integer greater than unity and $z$ is a complex number satisfying the equation $z^n = (z + 1)^n$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module