Question

The greatest positive argument of a complex number satisfying \[\left| {z - 4} \right| = \operatorname{Re} (z)\] is
A. \[\dfrac{\pi }{3}\]
B. \[\dfrac{{2\pi }}{3}\]
C. \[\dfrac{\pi }{2}\]
D. \[\dfrac{\pi }{4}\]

Answer 1

Hint: Knowing the basic concept that let the complex number be \[z = x + iy\]. Substitute it in the above equation and take modulus as \[z = \sqrt {{x^2} + {y^2}} \]and hence equate it with the real part of \[z\]. Hence, the argument of a complex number can be given as \[\arg .z = \tan \dfrac{y}{x}\]. Hence, substituting above values the required work can be done.

Complete step by step answer:

The given equation is \[\left| {z - 4} \right| = \operatorname{Re} (z)\] and so substitute the value of \[z = x + iy\]in above equation as
\[\left| {x + iy - 4} \right| = \operatorname{Re} (x + iy)\]
Hence, on simplifying
\[ \Rightarrow \]\[\left| {x - 4 + iy} \right| = x\]
Now take the modulus of L.H.S as,
\[ \Rightarrow \]\[\sqrt {{{\left( {x - 4} \right)}^2} + {y^2}} = x\]
On squaring both side
\[ \Rightarrow \]\[{\left( {\sqrt {{{\left( {x - 4} \right)}^2} + {y^2}} } \right)^2} = {x^2}\]
On simplifying the terms inside the bracket also and so,
\[ \Rightarrow \]\[{x^2} - 8x + 16 + {y^2} = {x^2}\]
On simplification,
\[ \Rightarrow \]\[{y^2} = 8(x - 2)\]
Hence, the given equation is of parabola
Compare it with the general equation and so the focus of parabola be \[\left( {4,0} \right)\].
Hence, the parabola drawn can have two possible tangents at a right angle to each other as,
Diagram:

Hence, the pair of the tangent from the directrix is at the right angle and so for a complex number, the possible highest argument can be
given as \[\dfrac{\pi }{4}\].
Hence, option(D) is the correct answer.

Note: A complex number is a number that can be expressed in the form \[a + ib\], where a and b are real numbers, and i represents the imaginary unit, satisfying the equation \[{i^2} =- 1\]. Because no real number satisfies this equation, i is called an imaginary number.