x-y-x-y-a-x-2-y-2-b-a-b-R-

Question Number 65054 by behi83417@gmail.com last updated on 24/Jul/19

{\begin{cases} \sqrt{x + y} + \sqrt{x - y} = a \\ x^{2} + y^{2} = b [a, b \in R] \end{cases}

Commented by behi83417@gmail.com last updated on 24/Jul/19

thanks in advance proph . Abdo .

[u + v = a \Rightarrow u^{2} + v^{2} + 2 uv = a^{2}] please check .

Commented by mathmax by abdo last updated on 24/Jul/19

y e s s i r \dots

Answered by MJS last updated on 24/Jul/19

x + y ⩾ 0 \Rightarrow x ⩾ - y

x - y ⩾ 0 \Rightarrow x ⩾ y

\Rightarrow 0 ⩽∣ y ∣⩽ x

\Rightarrow a ⩾ 0; b ⩾ 0

\sqrt{x + y} + \sqrt{x - y} = a \Rightarrow x = \frac{y^{2}}{a^{2}} + \frac{a^{2}}{4}

\Rightarrow a \neq 0

x^{2} + y^{2} = b

y^{4} + \frac{3 a^{4}}{2} y^{2} + \frac{a^{4} (a^{4} - 16 b)}{16} = 0

y^{2} = - \frac{3 a^{4}}{4} \pm \frac{a^{2} \sqrt{2 a^{4} + 4 b}}{2}

y^{2} ⩾ 0 \Rightarrow y^{2} = - \frac{3 a^{4}}{4} + \frac{a^{2} \sqrt{2 a^{4} + 4 b}}{2}

y = \pm \frac{a}{2} \sqrt{- 3 a^{2} + 2 \sqrt{2 a^{4} + 4 b}}

\Rightarrow x = - \frac{a^{2}}{2} + \frac{1}{2} \sqrt{2 a^{4} + 4 b}

Commented by MJS last updated on 24/Jul/19

if we allow y \in C

\Rightarrow y = \pm \frac{a}{2} \sqrt{- 3 a^{2} + 2 \sqrt{2 a^{4} + 4 b}} \lor y = \pm i \frac{a}{2} \sqrt{3 a^{2} + 2 \sqrt{2 a^{4} + 4 b}}

\Rightarrow x = - \frac{a^{2}}{2} + \frac{1}{2} \sqrt{2 a^{4} + 4 b} \lor x = - \frac{a^{2}}{2} - \frac{1}{2} \sqrt{2 a^{4} + 4 b}

Commented by MJS last updated on 24/Jul/19

{you}^{'} re welcome, and thank you

Commented by behi83417@gmail.com last updated on 24/Jul/19

thank you very much proph : MJS .

you have a great place in this forum and

also for me . god bless you sir .

Answered by mr W last updated on 24/Jul/19

x + y = p

x - y = q

\Rightarrow \sqrt{p} + \sqrt{q} = a > 0

{(x + y)}^{2} + {(x - y)}^{2} = 2 (x^{2} + y^{2})

\Rightarrow p^{2} + q^{2} = 2 b

l e t P = \sqrt{p}, Q = \sqrt{q}

P + Q = a \Rightarrow Q = a - P

P^{4} + Q^{4} = 2 b \Rightarrow P^{4} + {(a - P)}^{4} = 2 b

\Rightarrow 2 P^{4} - 4 {a P}^{3} + 6 a^{2} P^{2} - 4 a^{3} P + a^{4} - 2 b = 0

l e t λ = \frac{P}{a}

\Rightarrow λ^{4} - 2 λ^{3} + 3 λ^{2} - 2 λ + (\frac{1}{2} - \frac{b}{a^{4}}) = 0

\Rightarrow λ^{4} - 2 λ^{3} + 3 λ^{2} - 2 λ + δ = 0

\Rightarrow λ_{1, 2} = \frac{1}{2} [1 \pm \sqrt{4 \sqrt{1 - δ} - 3}]

\Rightarrow λ_{2, 3} = \frac{1}{2} [1 \pm i \sqrt{4 \sqrt{1 - δ} + 3}]

P_{1, 2} = a λ = \frac{a}{2} [1 \pm \sqrt{4 \sqrt{1 - δ} - 3}]

Q_{1, 2} = a - \frac{a}{2} [1 \pm \sqrt{4 \sqrt{1 - δ} - 3}] = \frac{a}{2} [1 \mp \sqrt{4 \sqrt{1 - δ} - 3}]

P_{3, 4} = a λ = \frac{a}{2} [1 \pm i \sqrt{4 \sqrt{1 - δ} + 3}]

Q_{3, 4} = a - \frac{a}{2} [1 \pm i \sqrt{4 \sqrt{1 - δ} + 3}] = \frac{a}{2} [1 \mp i \sqrt{4 \sqrt{1 - δ} + 3}]

x = \frac{p + q}{2} = \frac{P^{2} + Q^{2}}{2}

y = \frac{p - q}{2} = \frac{P^{2} - Q^{2}}{2}

\Rightarrow x_{1, 2} = \frac{a^{2}}{8} {{[1 \pm \sqrt{4 \sqrt{1 - δ} - 3}]}^{2} + {[1 \mp \sqrt{4 \sqrt{1 - δ} - 3}]}^{2}}

\Rightarrow x_{1, 2} = \frac{1}{2} a^{2} (2 \sqrt{1 - δ} - 1)

\Rightarrow x_{1, 2} = \frac{1}{2} [\sqrt{2 (a^{4} + 2 b)} - a^{2}]

\Rightarrow x_{3, 4} = \frac{a^{2}}{8} {{[1 \pm i \sqrt{4 \sqrt{1 - δ} + 3}]}^{2} + {[1 \mp i \sqrt{4 \sqrt{1 - δ} + 3}]}^{2}}

\Rightarrow x_{3, 4} = - \frac{a^{2}}{2} (2 \sqrt{1 - δ} + 1)

\Rightarrow x_{3, 4} = - \frac{1}{2} [\sqrt{2 (a^{4} + 2 b)} + a^{2}]

\Rightarrow y_{1, 2} = \frac{a^{2}}{8} {{[1 \pm \sqrt{4 \sqrt{1 - δ} - 3}]}^{2} - {[1 \mp \sqrt{4 \sqrt{1 - δ} - 3}]}^{2}}

\Rightarrow y_{1, 2} = \pm \frac{a^{2}}{2} \sqrt{4 \sqrt{1 - δ} - 3}

\Rightarrow y_{1, 2} = \pm \frac{a}{2} \sqrt{2 \sqrt{2 (a^{4} + 2 b)} - 3 a^{2}}

\Rightarrow y_{3, 4} = \frac{a^{2}}{8} {{[1 \pm i \sqrt{4 \sqrt{1 - δ} + 3}]}^{2} - {[1 \mp i \sqrt{4 \sqrt{1 - δ} + 3}]}^{2}}

\Rightarrow y_{3, 4} = \pm \frac{a^{2} i}{2} \sqrt{4 \sqrt{1 - δ} + 3}

\Rightarrow y_{3, 4} = \pm \frac{a i}{2} \sqrt{2 \sqrt{2 (a^{4} + 2 b)} + 3 a^{2}}

Commented by behi83417@gmail.com last updated on 24/Jul/19

my master is coming!

thanks in advance dear master .

your answers {don}^{'} t match with sir {MJS}^{'} s .

where is the problem sir ?

Commented by mr W last updated on 25/Jul/19

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

i n c l u d e s t h e c o m p l e x r o o t s .

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