Q.Distance of the point $(\alpha, \beta, \gamma)$ from $y$-axis is
(A) $\beta$
(B) $|\beta|$
(C) $|\beta| + |\gamma|$
(D) $\sqrt{\alpha^2 + \gamma^2}$

Concept: Distance From Y Axis — the distance of a point from the $y$-axis is the perpendicular distance to the $y$-axis, which lies in the $xz$-plane.

Reasoning:

1. The $y$-axis consists of all points $(0, y, 0)$. The perpendicular from $(\alpha, \beta, \gamma)$ to the $y$-axis lands at $(0, \beta, 0)$. …

The key idea: distance from a point to an axis is not the same as the coordinate value along that axis. Many students mistakenly think the distance from the y-axis is simply |β|, but that’s the distance from the xz-plane, not the axis.

Think of the y-axis as a vertical line through the origin. Any point’s distance to this line is measured perpendicularly — meaning we ignore the y-coordinate entirely. Why? Because moving up or down along the y-axis doesn’t change how far you are from the axis itself; only your horizontal (x and z) position matters.

1. Visualize the geometry. The y-axis consists of all points where x = 0 and z = 0. So the point (α, β, γ) is at a horizontal offset from this line. The y-coordinate β tells you how high the point is, but that’s parallel to the axis, not perpendicular.

2. Apply the distance formula in 3D. The distance from a point (x₁, y₁, z₁) to a line through the origin along the y-direction is the length of the component perpendicular to that direction. For the y-axis, the perpendicular components are the x and z coordinates.

3. Compute the perpendicular distance. Using the Pythagorean theorem in the xz-plane:

   $\text{Distance} = \sqrt{(\alpha - 0)^2 + (\gamma - 0)^2} = \sqrt{\alpha^2 + \gamma^2}$ …

Method: Distance of a Point from a Coordinate Axis

To find how far a point $(\alpha,\beta,\gamma)$ is from a coordinate axis, drop a perpendicular onto that axis — the coordinate that runs along the axis never contributes.

Steps

Step 1: Identify the coordinate that runs along the axis.

Points on the $y$-axis look like $(0,t,0)$, so moving along it changes only the $y$-value. That coordinate is parallel to the axis and plays no part in the distance.

Step 2: Keep the two perpendicular coordinates.

The remaining coordinates measure how far off the axis the point sits. For the $y$-axis those are $x$ and $z$.

Step 3: Combine them with the Pythagorean rule. …

Common Mistakes

Mistake 1: Answering $|\beta|$ (the $y$-coordinate) for the distance from the $y$-axis.

Why it's wrong: $|\beta|$ is the distance from the $xz$-plane, not from the $y$-axis; the $y$-coordinate runs along the axis and cannot measure distance from it. Correct approach: drop the coordinate that names the axis and combine the other two, giving $\sqrt{\alpha^2+\gamma^2}$.

Mistake 2: Adding the perpendicular coordinates instead of using Pythagoras. …