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Question Number 209520 by Ismoiljon_008 last updated on 12/Jul/24

A t w h a t v a l u e o f a d o e s t h e s y s t e m

o f e q u a t i o n s h a v e 4 s o l u t i o n s ?

{x^{2} - y^{2} = 0

{{(x - a)}^{2} + y^{2} = 1

Commented by Ismoiljon_008 last updated on 12/Jul/24

h e l p p l e a s e

Answered by Rasheed.Sindhi last updated on 12/Jul/24

{\begin{cases} x^{2} - y^{2} = 0 \Rightarrow x^{2} = y^{2} \\ {(x - a)}^{2} + y^{2} = 1 \end{cases}

x^{2} - 2 a x + a^{2} + y^{2} = 1

x^{2} - 2 a x + a^{2} + x^{2} = 1

2 x^{2} - 2 a x + a^{2} - 1 = 0

x = \frac{2 a \pm \sqrt{4 a^{2} - 4 (2) (a^{2} - 1)}}{4}

x = \frac{2 a \pm 2 \sqrt{2 - a^{2}}}{4}

x = \frac{a \pm \sqrt{2 - a^{2}}}{2} \Rightarrow 2 - a^{2} \neq 0 \Rightarrow a \neq \pm \sqrt{2}

F o r a l l o t h e r v a l u e s o f a

x h a s t w o v a l u e s

y^{2} = x^{2} \Rightarrow y = \pm ∣ x ∣

(\frac{a + \sqrt{2 - a^{2}}}{2}, \pm \frac{a + \sqrt{2 - a^{2}}}{2})

(\frac{a - \sqrt{2 - a^{2}}}{2}, \pm \frac{a - \sqrt{2 - a^{2}}}{2})

a \in R - {\pm \sqrt{2}}

Commented by Ismoiljon_008 last updated on 12/Jul/24

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