what-is-asymtote-og-function-y-2-x-2a-x-3-a-3-

Question Number 81195 by jagoll last updated on 10/Feb/20

w h a t i s a s y m t o t e o g f u n c t i o n

y^{2} (x - 2 a) = x^{3} - a^{3} ?

Commented by mr W last updated on 10/Feb/20

m y m e t h o d :

y^{2} (x - 2 a) = x^{3} - a^{3}

f (x) = y = \pm \sqrt{\frac{x^{3} - a^{3}}{x - 2 a}}

s a y t h e a s y m t o t e f o r f (x) = \sqrt{\frac{x^{3} - a^{3}}{x - 2 a}} i s

y = p x + q

f (x) - (p x + q) = \sqrt{\frac{x^{3} - a^{3}}{x - 2 a}} - p x - q

l e t t = x - 2 a

x = t + 2 a

x^{3} = t^{3} + 6 {a t}^{2} + 12 a^{2} t + 8 a^{3}

x^{3} - a^{3} = t^{3} + 6 {a t}^{2} + 12 a^{2} t + 7 a^{3}

\frac{x^{3} - a^{3}}{x - 2 a} = t^{2} (1 + \frac{6 a}{t} + \frac{12 a^{2}}{t^{2}} + \frac{7 a^{3}}{t^{3}})

\lim_{x \to \infty} (\sqrt{\frac{x^{3} - a^{3}}{x - 2 a}} - p x - q) = 0

\lim_{t \to \infty} [t \sqrt{1 + \frac{6 a}{t} + \frac{12 a^{2}}{t^{2}} + \frac{7 a^{3}}{t^{3}}} - p t - 2 p a - q] = 0

\lim_{t \to \infty} {t [1 + \frac{3 a}{t} + o (\frac{1}{t})] - p t - 2 p a - q} = 0

\Rightarrow p = 1

\Rightarrow 3 a - 2 p a - q = 0 \Rightarrow q = a

i . e . t h e a s y m y o t e f o r f (x) = \sqrt{\frac{x^{3} - a^{3}}{x - 2 a}}

i s y = x + a

s i m i l a r l y t h e a s y m y o t e f o r f (x) = - \sqrt{\frac{x^{3} - a^{3}}{x - 2 a}}

i s y = - x - a

\Rightarrow t h e a s y m t o t e s f o r y^{2} (x - 2 a) = x^{3} - a^{3} a r e

y = x + a a n d y = - x - a

Commented by mr W last updated on 10/Feb/20

Commented by jagoll last updated on 10/Feb/20

s i r v e r t i c a l a s y m t o t e i s x = 2 a

i t c o r r e c t ?

Commented by jagoll last updated on 10/Feb/20

t h a n k y o u v e r y m u c h s i r

Commented by ajfour last updated on 10/Feb/20

y = 0 a t x = a, x = 2 a i s v e r t i c a l

a s y m t o t e . I n y o u r d i a g r a m, S i r,

i t s e e m s a = 2 a! ?

Commented by mr W last updated on 10/Feb/20

Commented by mr W last updated on 10/Feb/20

t h i s i s a n e x a m p l e w i t h a = 5 .

v e r t i c a l a s y m t o t e i s x = 2 a = 10 .

x = 5 a t y = 0 .

Facebook Tweet Pin