Find the real numbers x and y if (x-iy) (3+5i) is the conjugate of -6-24i.
Solution and Explanation
\(Let z=(x-iy)(3+5i)\)
\(z=3x+5xi-3yi-5yi^2=3x+5xi-3yi+5y=(3x+5y)+i(5x-3y)\)
\(∴\bar{z}=(3x+5y)-i(5x-3y)=-6-24i\)
Equating real and imaginary parts, we obtain
3x+5y=-6….(i)
5x-3y=24……(ii)
Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain
\(9x+15y=-18\)
\(\frac{25x-15y=120}{34x=102}\)
\(∴ x=\frac{102}{34}=3\)
Putting the value of x in equation (i), we obtain
\(3(3)+5y=-6\)
\(⇒5y=-6-9=-15\)
\(⇒y=-3\)
Thus, the values of x and y are 3 and -3 respectively.
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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.