(A) The equation of a circle with centre $z_0$ and radius $r$ is given by $|z - z_0| = r$.Squaring both sides,we get $|z - z_0|^2 = r^2$.Using the pro

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$z\bar{z} - z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0} = r^2$

(A) The equation of a circle with centre $z_0$ and radius $r$ is given by $|z - z_0| = r$.Squaring both sides,we get $|z - z_0|^2 = r^2$.Using the property $|w|^2 = w\bar{w}$,we have $(z - z_0)(\overline{z - z_0}) = r^2$.$(z - z_0)(\bar{z} - \bar{z_0}) = r^2$.Expanding the product,we get $z\bar{z} - z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0} = r^2$.

Class 11 Mathematics 4-1.Complex numbers Geometry of complex numbers

Medium

$A$ circle whose radius is $r$ and centre is $z_0$,then the equation of the circle is

A
$z\bar{z} - z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0} = r^2$
B
$z\bar{z} + z\bar{z_0} - \bar{z}z_0 + z_0\bar{z_0} = r^2$
C
$z\bar{z} - z\bar{z_0} + \bar{z}z_0 - z_0\bar{z_0} = r^2$
D
None of these

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4-1.Complex numbers

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Geometry of complex numbers

Let $O$ be the origin and $A$ be the point $z_{1} = 1 + 2i$. If $B$ is the point $z_{2}$ with $\operatorname{Re}(z_{2}) < 0$, such that $\triangle OAB$ is a right-angled isosceles triangle with $OB$ as the hypotenuse, then which of the following is $NOT$ true?

DifficultJEE MAIN 2022

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The equation $\text{Re}(z^2) = 1$ represents which of the following?

Easy

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The set of points on the Argand plane which satisfy both $|z| \leq 4$ and $\operatorname{Arg}(z) = \frac{\pi}{3}$ represents:

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Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1| = 9$ and $|z_2 - (3 + 4i)| = 4$. Then the minimum value of $|z_1 - z_2|$ is

DifficultJEE MAIN 2019

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The solutions of the equation in $z$,$|z|^2 - (z + \bar{z}) + i(z - \bar{z}) + 2 = 0$ are $(i = \sqrt{-1})$.

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$A$ circle whose radius is $r$ and centre is $z_0$,then the equation of the circle is

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Let $O$ be the origin and $A$ be the point $z_{1} = 1 + 2i$. If $B$ is the point $z_{2}$ with $\operatorname{Re}(z_{2}) < 0$, such that $\triangle OAB$ is a right-angled isosceles triangle with $OB$ as the hypotenuse, then which of the following is $NOT$ true?

The equation $\text{Re}(z^2) = 1$ represents which of the following?

The set of points on the Argand plane which satisfy both $|z| \leq 4$ and $\operatorname{Arg}(z) = \frac{\pi}{3}$ represents:

Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1| = 9$ and $|z_2 - (3 + 4i)| = 4$. Then the minimum value of $|z_1 - z_2|$ is

The solutions of the equation in $z$,$|z|^2 - (z + \bar{z}) + i(z - \bar{z}) + 2 = 0$ are $(i = \sqrt{-1})$.

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