The perpendicular distance of the point `(6,5,8)` from y-axis is

Options

A. 5 units
B. 6 units
C. 8 units
D. 10 units

Explanation

To find the perpendicular distance of the point \( (6, 5, 8) \) from the y-axis, we can follow these steps:

### Step 1: Understand the coordinates of the point and the y-axis
The point given is \( (6, 5, 8) \). The y-axis consists of all points where the x-coordinate and z-coordinate are zero. Therefore, any point on the y-axis can be represented as \( (0, y, 0) \).

### Step 2: Identify the coordinates of the point on the y-axis
To find the perpendicular distance from the point \( (6, 5, 8) \) to the y-axis, we need to consider the corresponding point on the y-axis that has the same y-coordinate as our point. This point will be \( (0, 5, 0) \).

### Step 3: Use the distance formula
The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in a three-dimensional space is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Here, we will use the points \( (6, 5, 8) \) and \( (0, 5, 0) \).

### Step 4: Substitute the coordinates into the distance formula
Using the points \( (6, 5, 8) \) and \( (0, 5, 0) \):
- \( x_1 = 6, y_1 = 5, z_1 = 8 \)
- \( x_2 = 0, y_2 = 5, z_2 = 0 \)

Substituting into the formula:
\[ d = \sqrt{(0 - 6)^2 + (5 - 5)^2 + (0 - 8)^2} \]

### Step 5: Simplify the expression
Calculating each term:
- \( (0 - 6)^2 = 36 \)
- \( (5 - 5)^2 = 0 \)
- \( (0 - 8)^2 = 64 \)

Now substituting back:
\[ d = \sqrt{36 + 0 + 64} = \sqrt{100} \]

### Step 6: Calculate the final distance
\[ d = 10 \]

Thus, the perpendicular distance of the point \( (6, 5, 8) \) from the y-axis is **10 units**.

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