Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Perpendicular distance of the point \((3,4,5)\) from the \(y\)-axis, is (A) \(\sqrt{34}\) (B) \(\sqrt{41}\) (C) 4 (D) 5

Step-by-Step Solution

Answer

(A) \(\sqrt{34}\)

Step 1: Understand the Problem

We need to find the perpendicular distance of the point \((3,4,5)\) from the \(y\)-axis. The perpendicular distance from a point to an axis is generally the distance that is orthogonal to the axis.

Step 2: Analyze the Coordinates

The coordinates \((3,4,5)\) represent the \(x\), \(y\), and \(z\) values of the point in 3-dimensional space. Since we want the distance from the \(y\)-axis, we focus on the other two coordinates, \(x\) and \(z\).

Step 3: Determine Perpendicular Distance Formula

For a point \((x, y, z)\) to the \(y\)-axis, the perpendicular distance simplifies to the distance in the 2D xz-plane, calculated using the formula: \(d = \sqrt{x^2 + z^2}\).

Step 4: Plug in Values

Substitute the \(x\)-coordinate (3) and the \(z\)-coordinate (5) into the formula: \[d = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}\]

Step 5: Find the Correct Option

Now, compare the calculated distance \(\sqrt{34}\) with the given options (A) \(\sqrt{34}\), (B) \(\sqrt{41}\), (C) 4, (D) 5. The correct answer matches option (A) \(\sqrt{34}\).

Key Concepts

Perpendicular Distance Formula

To find the perpendicular distance from a point to an axis in 3-dimensional space, we use a special formula. This formula helps us see how far away a point is in a straight line from the axis we're interested in. For the perpendicular distance from a point \(x, y, z\) to the \(y\)-axis, you ignore the \(y\)-value. Instead, you focus on the \(x\) and \(z\) coordinates because the distance is measured in the xz-plane.

The formula for this perpendicular distance is given by:

• \( d = \sqrt{x^2 + z^2} \).

This formula calculates the straight-line distance from the point to the closest point on the \(y\)-axis. Essentially, what it does is use the Pythagorean theorem to determine the distance by treating the axes as perpendicular to each other, as it would in a 2-dimensional right-angled triangle.

3D Coordinate Geometry

In 3D coordinate geometry, every point in space is represented by three numbers: \(x, y, z\). These coordinates are essential because they help us locate any point in the three-dimensional world. Each of these numbers tells us how far along an axis the point is:

• The \(x\)-coordinate shows the horizontal distance from the \(yz\)-plane.
• The \(y\)-coordinate indicates the vertical distance from the \(xz\)-plane.
• The \(z\)-coordinate marks the depth from the \(xy\)-plane.

A point like \( (3, 4, 5) \) tells us that you move 3 units along the x-axis, then 4 units up the y-axis, and finally 5 units along the z-axis to get to that point in space.

Understanding these coordinates makes working with 3D models, virtual environments, and real-world problems easier. It's the basic language of space, and helps when calculating distances or angles in three dimensions.

Distance from an Axis

When finding the distance from a point to an axis—such as the \(y\)-axis in our exercise—it's important to understand the axis' role. In 3D geometry, an axis acts as a reference line. Measuring distance from an axis means finding how far a point is, perpendicular to that reference line.

• For instance, from the \(y\)-axis, the distance involves the other two coordinates, \(x\) and \(z\).

The concept is similar to how you might measure how far away someone stands next to a straight track or line. You don't count parallel distances along the line; you measure directly across to it.

In our example, using the coordinates \(3, 4, 5\), to find how far it is from the \(y\)-axis, you calculate as if math worked within a flat sheet or plane that ignores the \(y\)-coordinate altogether. Instead, it uses the horizontal and depth measurements provided by \(x\) and \(z\). This gives you a clear, perpendicular measure right to the axis.