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-

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 \dots T h a n k s s i r

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