Question

Выразить х через у. $y=2+sqrt{2x-x^{2} }$

Answer 1

$y = 2 + sqrt{2x - {x}^{2} } \ sqrt{2x - {x}^{2} } = y - 2 \ 2x - {x}^{2} = (y - 2)^{2} \ {x}^{2} - 2x + (y - 2)^{2} = 0 \ D = {2}^{2} - 4(y - 2)^{2} = 4 - 4( {y}^{2} - 4y + 4) = \ = 4 - 4 {y}^{2} + 16y - 16 = - 4 {y}^{2} + 16y - 12 = \ = - 4( {y}^{2} - 4y + 3) = - 4(y - 1)(y - 3) \ x_{1} = frac{2 - sqrt{ - 4(y - 1)(y - 3)} }{2} = 1 - sqrt{(1 - y)(y - 3)} \ x_{2} = frac{2 + sqrt{ - 4(y - 1)(y - 3)} }{2} = 1 + sqrt{(1 - y)(y - 3)}$