the number of ordered pairs (x,y) of real numbers satisfying 4x^2 −4x+2=sin^2 y and x^2 +y^2 ≤ 3 is ?


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Question Number 29478 by math solver last updated on 09/Feb/18

$t h e n u m b e r o f o r d e r e d p a i r s (x, y)$ $o f r e a l n u m b e r s s a t i s f y i n g$ $4 x^{2} - 4 x + 2 = {s i n}^{2} y$ $a n d x^{2} + y^{2} ⩽ 3 i s ?$
	Answered by mrW2 last updated on 09/Feb/18

	$4 (x^{2} - x + \frac{1}{4}) + 1 = {s i n}^{2} y$ $4 {(x - \frac{1}{2})}^{2} + 1 = {s i n}^{2} y$ $L H S ⩾ 1$ $R H S ⩽ 1$ $\Rightarrow L H S = R H S = 1$ $\Rightarrow x - \frac{1}{2} = 0 \Rightarrow x = \frac{1}{2}$ $\Rightarrow \sin y = \pm 1 \Rightarrow y = 2 n π \pm \frac{π}{2}$ $\frac{1}{4} + y^{2} ⩽ 3$ $\Rightarrow y^{2} < \frac{11}{4}$ $\Rightarrow - \frac{\sqrt{11}}{2} < y < \frac{\sqrt{11}}{2}$ $\Rightarrow y = \pm \frac{π}{2}$ $\Rightarrow (x, y) = (\frac{1}{2}, - \frac{π}{2}) a n d (\frac{1}{2}, \frac{π}{2})$
	Commented by Rasheed.Sindhi last updated on 09/Feb/18
	w♡w мяω2!
	Commented by math solver last updated on 09/Feb/18

	$t h a n k y o u s i r .$
	Commented by NECx last updated on 09/Feb/18

	$w o w . . . T h a n k s s i r$
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