A line is parallel to x-axis if all the points on the line have equal ____.

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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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In three dimensions a line parallel to the $x$ -axis runs in the $x$ -direction only, so every point on it has the same $y$ -coordinate and the same $z$ -coordinate; only $x$ changes.

Reasoning in Space

A point in space is $(x,y,z)$ . The $x$ -axis points purely in the $x$ -direction — along it, $y$ and $z$ stay $0$ .

If a line is parallel to the $x$ -axis, it points in that same pure $x$ -direction. As you travel along the line:

The $x$ -coordinate is free to increase or decrease — that is the direction of travel.
The perpendicular position of the line, measured across the $y$ -direction, cannot change — otherwise the line would tilt away from the $x$ -axis. So $y$ is constant.
Likewise the position across the $z$ -direction cannot change, so $z$ is constant.

Therefore both $y$ and $z$ are the same for every point on the line, while $x$ varies. Such a line is described by …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

TG EAPCET 2023Set eng-2023-05-12-FN1 markMCQ
$(x, y, z)$
›Reveal solutionSolution
The sum of the squares of perpendicular distances from a point to the coordinate axes equals twice the square of the distance from the origin, so $k = 2$ .

We are asked: For a point $(x, y, z)$ , the sum of the squares of its perpendicular distances to the three coordinate axes equals $k$ times the square of its distance from the origin. Find $k$ .

Concept & Intuition

The "perpendicular distance" from a point to an axis is the shortest distance from the point to any point on that axis. For the $x$ -axis, the foot of the perpendicular is $(x, 0, 0)$ , so the distance is $\sqrt{y^2 + z^2}$ . Similarly for the $y$ -axis and $z$ -axis. The distance from the origin is $\sqrt{x^2 + y^2 + z^2}$ . The problem asks how these quantities relate — specifically, the sum of the squares of those three axis-distances compared to the square of the origin-distance.

Step-by-step reasoning

Write the perpendicular distances to each axis

Distance to $x$ -axis: $\sqrt{y^2 + z^2}$

Distance to $y$ -axis: $\sqrt{x^2 + z^2}$

Distance to $z$ -axis: $\sqrt{x^2 + y^2}$

Square each distance

Square of distance to $x$ -axis: $y^2 + z^2$

Square of distance to $y$ -axis: $x^2 + z^2$

Square of distance to $z$ -axis: $x^2 + y^2$

Sum these squares

$(y^2 + z^2) + (x^2 + z^2) + (x^2 + y^2) = 2x^2 + 2y^2 + 2z^2$

Compare with the square of the distance from the origin …
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Q.A line is parallel to xxx-axis if all the points on the line have equal ________.

Reasoning in Space

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Q.A line is parallel to $x$ -axis if all the points on the line have equal ________.

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