Equation of y-axis is considered as (A) x=0,y=0 (B) y=0,z=0 (C) z=0,x=0 (D) none of these

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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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The $y$ -axis is the set of all points where both $x$ and $z$ coordinates are zero, while $y$ varies freely. The equation is $x = 0, z = 0$ .

Understanding Coordinate Axes in Three Dimensions

When we work in three-dimensional space, each axis is defined by fixing the other two coordinates at zero. Think of it this way: the $y$ -axis is the vertical line (in standard orientation) that passes through the origin and extends in both directions along the $y$ -direction only.

A point lies on the $y$ -axis if and only if it has no displacement in the $x$ -direction and no displacement in the $z$ -direction. Such a point has coordinates of the form $(0, y, 0)$ where $y$ can be any real number.

Step-by-Step Reasoning

Recall the general form of a point in 3D space

Any point in three-dimensional Cartesian space is written as $(x, y, z)$ .
Identify what makes a point lie on the $y$ -axis

For a point to be on the $y$ -axis, it must have:
- No $x$ -component: $x = 0$
- No $z$ -component: $z = 0$
- An arbitrary $y$ -component: $y \in \mathbb{R}$
Write the equation of the $y$ -axis …

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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.Equation of yyy-axis is considered as (A) x=0, y=0x=0,\ y=0x=0, y=0 (B) y=0, z=0y=0,\ z=0y=0, z=0 (C) z=0, x=0z=0,\ x=0z=0, x=0 (D) none of these

Understanding Coordinate Axes in Three Dimensions

Step-by-Step Reasoning

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Q.Equation of $y$ -axis is considered as
(A) $x=0,\ y=0$
(B) $y=0,\ z=0$
(C) $z=0,\ x=0$
(D) none of these

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