If the point P lies on z-axis, then coordinates of P are of the form ____.

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Concept understanding — Coordinate Axes Equations

Coordinate Axes Equations

Imagine a blank sheet of paper. You want to describe exactly where a point is. The most natural way is to draw two perpendicular lines — one horizontal, one vertical — and measure distances from them. Those two lines are the coordinate axes.

The horizontal line is the x-axis. The vertical line is the y-axis. Where they cross is the origin, the point $(0,0)$ .

Now, here's the key idea: every point on the x-axis has a y-coordinate of 0. Why? Because to be on that horizontal line, you haven't moved up or down at all — your vertical distance from the axis is zero. Similarly, every point on the y-axis has an x-coordinate of 0 — you haven't moved left or right from the vertical line.

That's the entire intuition. Let's make it precise.

The Equations

\text{x-axis: } y = 0

That's it. These are the equations of the coordinate axes.

$y = 0$ means: "all points where the y-coordinate is zero, regardless of x." That's the entire horizontal line through the origin.
$x = 0$ means: "all points where the x-coordinate is zero, regardless of y." That's the entire vertical line through the origin.

Watch out

A common mistake is to think $y=0$ means "the point $(0,0)$ ". No — it means every point with $y=0$ , like $(5,0)$ , $(-3,0)$ , $(100,0)$ , etc. It's a whole line, not a single point.

Why This Matters

These two equations are the foundation for everything in coordinate geometry. Every other line, curve, or shape is described relative to these axes. For example:

The line $y = 2$ is a horizontal line parallel to the x-axis, shifted up by 2 units.
The line $x = -3$ is a vertical line parallel to the y-axis, shifted left by 3 units.

So when you see $y = 0$ or $x = 0$ , think: "the original, unshifted axes themselves." …

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A point lying on the $z$ -axis has its $x$ and $y$ coordinates equal to zero, so its coordinates are of the form $\boxed{(0, 0, k)}$ .

In three-dimensional geometry, the position of any point in space is uniquely determined by an ordered triplet of real numbers $(x, y, z)$ . These numbers represent the perpendicular distances of the point from three mutually perpendicular planes, which intersect at a common point called the origin $(0, 0, 0)$ .

Specifically:

The $x$ -coordinate represents the perpendicular distance of the point from the $YZ$ -plane.
The $y$ -coordinate represents the perpendicular distance of the point from the $XZ$ -plane.
The $z$ -coordinate represents the perpendicular distance of the point from the $XY$ -plane.

When we say a point lies on the $z$ -axis, we are describing a very specific location. Imagine the $z$ -axis as a straight line passing through the origin and extending infinitely in both positive and negative $z$ directions. For a point to be on this line, it cannot have any displacement in the $x$ or $y$ directions from the origin.

Let's break this down step-by-step.

Understanding the Coordinate Axes:

The $x$ -axis is the line where $y=0$ and $z=0$ .

The $y$ -axis is the line where $x=0$ and $z=0$ .

The $z$ -axis is the line where $x=0$ and $y=0$ .

The origin is the point $(0,0,0)$ , where all three axes intersect.
Condition for Lying on the $z$ -axis:

If a point P lies on the $z$ -axis, it means that its position is solely determined by its distance along the $z$ -axis from the origin. It has no "sideways" displacement in the $x$ or $y$ directions.

This implies that the perpendicular distance of P from the $YZ$ -plane must be zero. Since the $x$ -coordinate represents this distance, we must have $x=0$ .

Similarly, the perpendicular distance of P from the $XZ$ -plane must also be zero. Since the $y$ -coordinate represents this distance, we must have $y=0$ .
Determining the $z$ -coordinate: …

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Q.If the point P lies on $z$ -axis, then coordinates of P are of the form ________.

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