4.1 Q-1

Question Statement

Describe the location in the plane of the point $P(x, y)$ based on the given conditions.

Each sub-part provides a specific condition for $x$ and/or $y$ to determine the region or exact placement of $P(x, y)$ in the coordinate plane.

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Background and Explanation

The Cartesian coordinate system divides the plane into four quadrants based on the signs of $x$ (abscissa) and $y$ (ordinate):

1. Quadrant I: $x > 0$ and $y > 0$
2. Quadrant II: $x < 0$ and $y > 0$
3. Quadrant III: $x < 0$ and $y < 0$
4. Quadrant IV: $x > 0$ and $y < 0$

Points on the axes do not belong to any quadrant but are located either on the x-axis or y-axis. Absolute values like $|x|$ and $|y|$ describe the magnitude of coordinates, regardless of sign.

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Solution

i. $x > 0$

• The condition $x > 0$ places the point in the right half-plane, which includes Quadrants I and IV.

ii. $x > 0$ and $y > 0$

• Both $x$ and $y$ are positive, so the point is in Quadrant I.

iii. $x = 0$

• When $x = 0$, the point lies on the y-axis, as there is no horizontal displacement.

iv. $y = 0$

• When $y = 0$, the point lies on the x-axis, as there is no vertical displacement.

v. $x < 0$ and $y \geq 0$

• Negative $x$ places the point in the left half-plane, and $y \geq 0$ restricts it to Quadrant II or on the positive y-axis.

vi. $x = y$

• For $x = y$, the point lies along the line $y = x$. Both coordinates are equal:
  • If $x > 0$, the point is in Quadrant I.
  • If $x < 0$, the point is in Quadrant III.

vii. $|x| = -|y|$

• Absolute values are always non-negative. Since $|x| = -|y|$ equates a positive value to a negative value, this is only possible at $(0, 0)$, the origin.

viii. $|x| > 3$

• $|x| > 3$ implies $x > 3$ or $x < -3$, so the point lies on the x-axis to the left of $-3$ or to the right of $3$.

ix. $x > 2$ and $y = 2$

• A fixed ordinate $y = 2$ and $x > 2$ place the point in Quadrant I with a vertical height of 2 and horizontal displacement greater than 2.

x. $x$ and $y$ have opposite signs

• Opposite signs mean:
  • $x > 0$ and $y < 0$: Point in Quadrant IV.
  • $x < 0$ and $y > 0$: Point in Quadrant II.

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Key Formulas or Methods Used

• Quadrant identification based on signs of $x$ and $y$.
• Absolute values to compare magnitudes regardless of sign.
• Axis determination for $x = 0$ or $y = 0$ conditions.

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Summary of Steps

1. Analyze the sign or equality constraints for $x$ and $y$.
2. Identify the quadrant, axis, or specific point based on the given condition.
3. Use key properties of absolute values and coordinate geometry when necessary.
4. Write the conclusion for each condition.