x-a-y-b-a-2-b-2-a-b-R-ab-x-2-y-2-xy-a-2-b-2-

Question Number 80145 by behi83417@gmail.com last updated on 31/Jan/20

{\begin{cases} \frac{x}{a} + \frac{y}{b} = a^{2} + b^{2} \\ [a, b \in R] \\ ab (x^{2} - y^{2}) = xy (a^{2} - b^{2}) \end{cases}

Commented by john santu last updated on 31/Jan/20

\frac{x^{2} - y^{2}}{x y} = \frac{a^{2} - b^{2}}{a b}

\frac{x}{y} - \frac{y}{x} = \frac{a}{b} - \frac{b}{a}

l e t \frac{x}{y} = t, \frac{a}{b} = p

t - \frac{1}{t} = p - \frac{1}{p} \Rightarrow t - p = \frac{1}{t} - \frac{1}{p}

t - p = - (\frac{t - p}{t p}) \Rightarrow (t - p) {1 + \frac{1}{t p}} = 0

(1) t = p \lor t p = - 1

Commented by john santu last updated on 31/Jan/20

\Rightarrow \frac{x}{a} + \frac{y}{b} = a^{2} + b^{2}

f o r t = p \Rightarrow \frac{b x + a y}{a b} = a^{2} + b^{2}

\frac{2 b x}{a b} = a^{2} + b^{2} \Rightarrow x = \frac{a}{2} (a^{2} + b^{2})

Commented by john santu last updated on 31/Jan/20

a y = b x \Rightarrow y = \frac{b}{a} \times \frac{a}{2} (a^{2} + b^{2})

y = \frac{b}{2} (a^{2} + b^{2})

★ d o n e b y s a n t u

Commented by jagoll last updated on 31/Jan/20

g r e a t \dots . m i s t e r

Commented by behi83417@gmail.com last updated on 31/Jan/20

thanks in advance sir : santu .

Answered by MJS last updated on 31/Jan/20

(1) \Rightarrow y = - \frac{b}{a} x + (a^{2} + b^{2}) b

\Rightarrow

(2) \dots

x^{2} - \frac{a (a^{2} + b^{2}) (a^{2} - 3 b^{2}) x}{2 (a^{2} - b^{2})} - \frac{a^{2} {(a^{2} + b^{2})}^{2} b^{2}}{2 (a^{2} - b^{2})} = 0

\Rightarrow

x = \frac{a (a^{2} + b^{2})}{2} \land y = \frac{(a^{2} + b^{2}) b}{2}

\lor

x = - \frac{a (a^{2} + b^{2}) b^{2}}{a^{2} - b^{2}} \land y = \frac{a^{2} (a^{2} + b^{2}) b}{a^{2} - b^{2}}

Commented by behi83417@gmail.com last updated on 31/Jan/20

thank you very much proph - : MJS

thank you very much proph - : MJS