Question

If $\arg \left( {z - a} \right) = \dfrac{{pi}}{4}$, where a belongs to R, then the locus of z belongs to c is a
A) Hyperloop
B) Parabola
C) Ellipse
D) Straight line

Answer 1

Hint: In this question a term argument is used. The argument of a complex number is an angle that is inclined from the real axis towards the direction of the complex number which is represented on a complex plane. Also, the locus term is used here. locus means a curve shape formed by all the points satisfying a specific equation of the relation between the coordinates, or by a point, line or moving surface.

Complete step by step answer:
In the given question, Let, $z = x + iy$
Given,
$\arg \left( {z - a} \right) = \dfrac{\pi }{4}$
Now,
Substitute the value of $z = x + iy$
$\arg \left( {x + iy - a} \right) = \dfrac{\pi }{4}$
We can also write,
$\arg \left( {\left( {x - a} \right) + iy} \right) = \dfrac{\pi }{4}$
As we know, the argument of a complex number tells us about the location of the complex number in the argand plane.
So,
${\tan ^{ - 1}}\left( {\dfrac{y}{{x - a}}} \right) = \dfrac{\pi }{4}$
Taking $\tan$ on both sides,
$\tan {\tan ^{ - 1}}\left( {\dfrac{y}{{x - a}}} \right) = \tan \dfrac{\pi }{4}$
As we know that, $\tan \left( {\dfrac{\pi }{4}} \right) = 1$
 $\dfrac{y}{{x - a}} = 1$
Using cross-multiplication,
$y = x - a$
$x - y = a$
Therefore, it represents a straight line. Hence, the correct option is option (D).

Note:
The argument of a complex number is defined as the angle inclined from the real axis in the
direction of the complex number represented on the complex plane. It is denoted on the complex plane. It is denoted by it is measured in the standard unit called radians. Locus of complex numbers is obtained by letting $\left( {z = x + iy} \right)$ and simplifying the expressions. Operations of modulus, conjugate pairs and arguments are to be used for determining the locus of complex numbers.