(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

$POQ$ is a straight line through the origin $O$. $P$ and $Q$ represent the complex numbers $z_1 = a + ib$ and $z_2 = c + id$ respectively. If $OP = OQ$,then:

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If $z = x + iy$ and the point $P$ represents $z$ in the Argand plane,then the locus of $z$ satisfying the equation $|z - 1| + |z + i| = 2$ is

MediumTS EAMCET 2024

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In the Argand plane, the distinct roots of $1+z+z^{3}+z^{4}=0$ ($z$ is a complex number) represent vertices of

MediumWBJEE 2014

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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

DifficultJEE MAIN 2022

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If the vertices of a quadrilateral are $A = 1 + 2i,$ $B = -3 + i,$ $C = -2 - 3i,$ and $D = 2 - 2i,$ then the quadrilateral is:

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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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$POQ$ is a straight line through the origin $O$. $P$ and $Q$ represent the complex numbers $z_1 = a + ib$ and $z_2 = c + id$ respectively. If $OP = OQ$,then:

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