find-the-complex-form-of-equation-4x-2-2y-2-5-

Question Number 113262 by bemath last updated on 12/Sep/20

find the complex form of

equation {4 x}^{2} - {2 y}^{2} = 5

Answered by mr W last updated on 12/Sep/20

\frac{x^{2}}{{(\frac{\sqrt{5}}{2})}^{2}} - \frac{y^{2}}{{(\frac{\sqrt{5}}{\sqrt{2}})}^{2}} = 1

c^{2} = a^{2} + b^{2} = \frac{5}{4} + \frac{5}{2} = \frac{15}{4}

c = \frac{\sqrt{15}}{2}

{i t}^{'} s a h y p e r b o l a w i t h f o c i (- \frac{\sqrt{15}}{2}, 0)

a n d (\frac{\sqrt{15}}{2}, 0) a s w e l l a s s e m i m a j o r

a x i s a = \frac{\sqrt{5}}{2} .

a c c o r d i n g t o t h e d e f i n i t i o n o f

h y p e r b o l a w e c a n w r i t e i t s e q u a t i o n

a l s o i n c o m p l e x f o r m :

∣ z + c ∣ - ∣ z - c ∣= \pm 2 a

o r

∣ z + \frac{\sqrt{15}}{2} ∣ - ∣ z - \frac{\sqrt{15}}{2} ∣= \pm \sqrt{5}

Commented by mr W last updated on 12/Sep/20

c e r t a i n l y y o u c a n a l s o w r i t e t h e

e q u a t i o n i n m a n y o t h e r c o m p l e x f o r m s .

Commented by bemath last updated on 12/Sep/20

thank you sir

Answered by john santu last updated on 12/Sep/20

l e t z = x + i y a n d \bar{z} = x - i y

{\begin{cases} z + \bar{z} = (2 x, 0); x = \frac{z + \bar{z}}{2} \\ z - \bar{z} = (0, 2 y); y = \frac{z - \bar{z}}{2} \end{cases}

\Leftrightarrow 4 x^{2} - 2 y^{2} = 5 i n c o m p l e x f o r m

4 {(\frac{z + \bar{z}}{2})}^{2} - 2 {(\frac{z - \bar{z}}{2})}^{2} = 5

{(z + \bar{z})}^{2} - \frac{{(z - \bar{z})}^{2}}{2} = 5

2 {(z + \bar{z})}^{2} - {(z - \bar{z})}^{2} = 10