Number of ordered pair(s) of (x, y) satisfying the system of simultaneous equations $∣ ∣ x^{2} - 2 x ∣ ∣ + y = 1 a n d x^{2} + | y | = 1 i s (x, y ϵ R)$

1

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2

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3

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infinitely many

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Solution

The correct option is C $3$
Given equations $| x^{2} - 2 x | + y = 1$ (1)
$x^{2} + | y | = 1$ (2)
Consider equation (1)
$| (x) (x - 2) | + y = 1$
For $x > 2, (x - 2) + y = 1$
For $0 < x < 2, - (x) (x - 2) + y = 1$
For $x < - 2, x (x - 2) + y = 1$
Consider equation (2)
For $y > 0, x^{2} + y = 1$
For $y < 0, x^{2} - y = 1$
By solving equations we get
$(x, y) = (1, 0), (- 1, 0), (0, 1)$

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