If \(a ≠ ±b\) and are purely real, z ϵ complex number, \(Re(az^{2}+bz)=a\) and \(Re(bz^{2}+az)=b\) then number of value of z possible is
- 0
- 1
- 2
- 3
The Correct Option is A
Approach Solution - 1
\(a(x^2-y^2)+bx=a…..(i)\)
\(b(x^2-y^2)+ax=b…..(ii)\)
\((i)-(ii)\)
\((a-b)(x^2-y^2)+(b-a)x=a-b\ \ \ \ (a≠b)\)
\(⇒ x^2-y^2-x=1\)
\((i)+(ii)\)
\((a+b)(x^2-y^2)+x(a+b)=a+b\) \( (a≠-b)\)
\(⇒ x^2-y^2+x=1\)
\(⇒x=0\)
\(⇒y^2=-1\)
therefore, no complex number is possible.
The correct option is (A): 0
Approach Solution -2
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Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
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Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.