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# If the amplitude of $z - 2 - 3i\;$ is $\dfrac{\pi }{4}$, then the locus of $z = x + iy$ is [EAMCET$2003$]A) $x + y - 1 = 0$B) $x - y - 1 = 0$C) $x + y + 1 = 0$D) $x - y + 1 = 0$

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Hint: in this question we have to find locus of the point $z$ which satisfy the given condition. First write the given complex number as a combination of real and imaginary number. Amplitude is same as argument. Then apply formula for argument.

Formula Used:Equation of complex number is given by
$z = x + iy$
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
i is iota
Square of iota is equal to the negative of one
$\arg (z) = {\tan ^{ - 1}}(\dfrac{y}{x})$

Complete step by step solution:Given: Amplitude of complex number is given
Now we have amplitude which is equal to$z - 2 - 3i\;$
We know that complex number is written as a combination of real and imaginary number.
$z = x + iy$
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
Put this value in$z - 2 - 3i\;$
$x + iy - 2 - 3i\;$
$x + iy - 2 - 3i\; = (x - 2) + i(y - 3)$
$\arg (z) = {\tan ^{ - 1}}(\dfrac{y}{x})$
It is given in the question that amplitude is equal to$\dfrac{\pi }{4}$
${\tan ^{ - 1}} = \dfrac{{y - 3}}{{x - 2}} = \dfrac{\pi }{4}$
$\tan \dfrac{\pi }{4} = \dfrac{{y - 3}}{{x - 2}}$
We know that
$\tan \dfrac{\pi }{4} = 1$
$\dfrac{{y - 3}}{{x - 2}} = 1$
$y - 3 = x - 2$
Now locus is given by
$x - y + 1 = 0$

Option ‘A’ is correct

Note: Here we have to remember that amplitude is equal to argument. Complex number is a number which is a combination of real and imaginary number. So in combination number question we have to represent number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to negative one.

Last updated date: 28th Sep 2023
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