Name: Basic Geometrical Concepts of Complex Number Plane
Uploaded: 2019-07-19
Duration: 5 min 49 s
Description: Before studying the geometrical applications of complex numbers, it is essential to recall the basic geometrical concepts of the complex number plane. Let's...

Question

Given

(x + i y)^{2} = 55 + 48 i

. Find the value of

x

and the value of

y

, when

x

and

y

are real and positive.

Accepted Answer

We have, $(x + i y)^{2} = 55 + 48 i$

$y$

$\Rightarrow x^{2} - y^{2} + 2 x y i = 55 + 48 i [∵ i^{2} = - 1]$

Equating real parts, we get:

$x^{2} - y^{2} = 55 . . . (1)$

Equating imaginary parts, we get:

$2 x y = 48 . . . (2)$

From equation $(2)$ , we get:

$y = \frac{24}{x} . . . (3)$

Putting the value of $y$ in equation $(1)$ , we get:

$x^{2} - {(\frac{24}{x})}^{2} = 55$

$\Rightarrow {(x^{2})}^{2} - 576 = 55 x^{2}$

$\Rightarrow {(x^{2})}^{2} - 55 x^{2} - 576 = 0$

$\Rightarrow x^{4} + 9 x^{2} - 64 x^{2} - 576 = 0$

$\Rightarrow x^{2} (x^{2} + 9) - 64 (x^{2} + 9) = 0$

$\Rightarrow (x^{2} - 64) (x^{2} + 9) = 0$

$\Rightarrow (x^{2} - 64) = 0 or (x^{2} + 9) = 0$

$\Rightarrow x^{2} = 64 or x^{2} = - 9$

$\Rightarrow x = \pm 8 or x = \pm \sqrt{- 9}$

$\Rightarrow x = \pm 8 or x = \pm 3 i$ $[∵ i^{2} = - 1]$

We are given that $x$ is real and positive, so $x = 8$ .

$\Rightarrow y = \frac{24}{8} = 3$

Thus, the required values are $x = 8$ and $y = 3$ .

Given $(x + i y)^{2} = 55 + 48 i$ . Find the value of $x$ and the value of $y$ , when $x$ and $y$ are real and positive.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given (x+iy)2=55+48i. Find the value of x and the value of y, when x and y are real and positive.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given $(x + i y)^{2} = 55 + 48 i$ . Find the value of $x$ and the value of $y$ , when $x$ and $y$ are real and positive.