a-x-2-y-2-b-x-y-c-amp-x-2-y-2-R-2-Solve-for-x-or-y-

Question Number 47966 by ajfour last updated on 17/Nov/18

a (x^{2} + y^{2}) + b (x + y) = c

& x^{2} - y^{2} = R^{2}

S o l v e f o r x o r y .

Commented by behi83417@gmail.com last updated on 17/Nov/18

x + y = t, x - y = s \Rightarrow 4 x y = t^{2} - s^{2}

a (t^{2} - \frac{t^{2} - s^{2}}{2}) + b t = c, t s = R^{2}

\Rightarrow a (t^{2} + s^{2}) + 2 b t = 2 c \Rightarrow a (t^{2} + \frac{R^{4}}{t^{2}}) + 2 b t = 2 c

\Rightarrow {a t}^{4} + 2 {b t}^{3} + 2 {c t}^{2} + {a R}^{4} = 0

\Rightarrow t^{4} + \frac{2 b}{a} t^{3} + \frac{2 c}{a} t^{2} + R^{4} = 0

b y h a v i n g n u m e r i c v a l v e s, w e c a n

s o l v e t h i s f o r t a n d t h e n s .

Answered by MJS last updated on 17/Nov/18

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

x + y = \frac{R^{2}}{x - y}

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

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

x = z + \frac{y}{3}

z^{3} + \frac{2 {a y}^{2} - 3 c}{3 a} z + \frac{27 {b R}^{2} - 2 y (10 {a y}^{2} - 9 c)}{27 a} = 0

and this can be solved \dots

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