Point has -coordinate equal to . Can you predict the coordinates of point which is on the line through parallel to the -axis? Which quadrants can lie in?

Question

Homi · Accepted Answer

Understand the Question

A line parallel to the $y$ -axis is a vertical line where every point shares the same $x$ -coordinate, represented by the equation $x = c$ .
Since the $x$ -coordinate of point $W$ is $-5$ , the line passing through $W$ has the equation $x = -5$ .
Any point $H$ on this line has coordinates of the form $(-5, y)$ , where $y$ can be any real number.
Because the $x$ -coordinate is negative, the point can only lie in Quadrant II (where $x < 0, y > 0$ ), Quadrant III (where $x < 0, y < 0$ ), or on the negative $x$ -axis (where $y = 0$ ).

Step 1 · Determine the Form of the Coordinates of $H$

A line parallel to the $y$ -axis has a constant $x$ -coordinate.Since the $x$ -coordinate of $W$ is $-5$ , the equation of the line passing through $W$ is $x = -5$

Every point on this line must have an $x$ -coordinate of $-5$ . The $y$ -coordinate cannot be uniquely determined without additional information, so the coordinates of $H$ are of the form $(-5, y) \quad \text{where } y \in \mathbb{R}$

Step 2 · Determine the Possible Quadrants for $H$

For point $H(-5, y)$ , the $x$ -coordinate is strictly negative ( $x = -5 < 0$ ):

If $y > 0$ , then $(-5, y)$ lies in Quadrant II ( $x < 0, y > 0$ ).
If $y < 0$ , then $(-5, y)$ lies in Quadrant III ( $x < 0, y < 0$ ).
If $y = 0$ , then $(-5, 0)$ lies on the negative $x$ -axis.

Therefore, point $H$ can lie in Quadrant II or Quadrant III.

Answer

Coordinates of $H$ are of the form $(-5, y)$ (where $y \in \mathbb{R}$ ). $H$ can lie in Quadrant II or Quadrant III (or on the negative $x$ -axis).

Common Mistakes

Assuming a Unique Coordinate: We cannot predict a single exact pair of coordinates because the $y$ -coordinate can be any real number along the vertical line $x = -5$ .
Confusing Axis Equations: Remembering that a line parallel to the $y$ -axis has the equation $x = c$ (vertical line), not $y = c$ (horizontal line).
Quadrant Signs: Misidentifying quadrants where $x < 0$ . Points with a negative $x$ -coordinate can only lie in Quadrants II and III, never Quadrants I or IV.

Question 2

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