GATE CS 2015 Set 3 QGA-8 — General Aptitude

To find the correct equation, we can analyze the behavior of the function $x$ in terms of $y$ by splitting the absolute value term $|y|$ into two cases: $y \ge 0$ and $y < 0$.

Let's evaluate option (B): $x = -(y - |y|)$

• Case 1: $y \ge 0$
  Here, $|y| = y$. Substituting this into the equation: $$x = -(y - y) = 0$$ This represents the positive $y$-axis (the vertical line segment where $x = 0$ for $y \ge 0$).
• Case 2: $y < 0$
  Here, $|y| = -y$. Substituting this into the equation: $$x = -(y - (-y)) = -2y$$ Since $y$ is negative, $x = -2y$ is positive. This represents a straight line in the fourth quadrant (positive $x$, negative $y$) starting at the origin.

This piecewise behavior (the positive $y$-axis combined with a line in the fourth quadrant) perfectly matches the thick line shown in the plot.

For comparison, the behavior of the other options is as follows:

• Option (A) $x = y - |y|$: For $y \ge 0$, $x = 0$. For $y < 0$, $x = 2y$ (extends into the third quadrant).
• Option (C) $x = y + |y|$: For $y \ge 0$, $x = 2y$ (extends into the first quadrant). For $y < 0$, $x = 0$.
• Option (D) $x = -(y + |y|)$: For $y \ge 0$, $x = -2y$ (extends into the second quadrant). For $y < 0$, $x = 0$.