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enter image description here According to me the right answer is option 2 as it satisfies the point (2,-1) also. But answer says it's option 1 which is the answer. Iam not convinced with the answer. Is my answer correct?

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    Your answer is correct, and is the only correct one among the three options. – 2017-02-04
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    @dxiv Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue? Seems to me like your comment answers the question pretty succinctly. – 2018-11-24
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    @RobertHoward Feel free to convert my comment into an answer, that would be entirely fair play here. – 2018-11-25

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As dxiv pointed out, your answer is correct. We can verify that by plugging $y=-1$ into the three equations given to see which equation yields $x=2$:

$$\begin{align} y-|y|&=-1-|-1|=-1-1=-2 \:\:\:\large{\times}\\[1ex] -(y-|y|)&=-(-1-1)=-(-2)=2 \qquad\checkmark \\[1ex] y+|y|&=-1+1=0 \hspace{3.4cm}\large\times \end{align}$$