For what values of a does the system of equations only have one solution: { ((a(x^4 +1)=y+2−∣x∣)),((x^2 +y^2 =4)) :}


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Question Number 181182 by depressiveshrek last updated on 22/Nov/22
$For what values of a does the system of equations only have one solution: { ((a(x^4 +1)=y+2−∣x∣)),((x^2 +y^2 =4)) :}$
$F o r w h a t v a l u e s o f a d o e s t h e s y s t e m$ $o f e q u a t i o n s o n l y h a v e o n e s o l u t i o n :$ ${\begin{cases} a (x^{4} + 1) = y + 2 - ∣ x ∣ \\ x^{2} + y^{2} = 4 \end{cases}$
	Answered by mr W last updated on 22/Nov/22

	$y = a (x^{4} + 1) + ∣ x ∣ - 2$ $l e t t =∣ x ∣> 0$ $y = {a t}^{4} + t + a - 2 . . . (i)$ $y^{2} + t^{2} = 2^{2} . . . (i i)$ $(i) a n d (i i) s h o u l d t o u c h a t o n e a n d$ $o n l y o n e p o i n t . d u e t o s y m m e t r y,$ $t h i s p o i n t m u s t b e a t t = 0, i . e . (0, 2)$ $a (0^{4} + 1) + 0 - 2 = 2$ $\Rightarrow a = 4 ✓$
	Commented by mr W last updated on 22/Nov/22

	Commented by depressiveshrek last updated on 22/Nov/22

	$T h a n k y o u s i r, y o u a r e h e l p i n g m e$ $a l o t . C a n I c o n t a c t y o u s o m e h o w ?$ $I n e e d h e l p t o p r a c t i c e f o r m y$ ${c o u n t r y}^{'} s n a t i o n a l m a t h o l y m p i a d .$
	Commented by mr W last updated on 22/Nov/22

	$h e r e i s a g o o d p l a c e . a l l p e o p l e c a n$ $h e l p y o u .$
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