Perpendicular distance of point (3, 4, 5) from Y-axis is

Options

A. 4
B. 5
C. `sqrt(34)`
D. `sqrt(41)`

Answer

To find the perpendicular distance of the point \( P(3, 4, 5) \) from the Y-axis, we can follow these steps:

### Step 1: Identify the coordinates of the point and the Y-axis
The point given is \( P(3, 4, 5) \). The Y-axis can be represented by the points where \( x = 0 \) and \( z = 0 \). Therefore, any point on the Y-axis can be represented as \( A(0, y, 0) \).

### Step 2: Determine the coordinates of the closest point on the Y-axis
The closest point on the Y-axis to the point \( P(3, 4, 5) \) will have the same \( y \)-coordinate as point \( P \) and \( x \) and \( z \) coordinates equal to 0. Thus, the coordinates of the point on the Y-axis are \( A(0, 4, 0) \).

### Step 3: Use the distance formula
The distance \( d \) between two points \( P(x_1, y_1, z_1) \) and \( A(x_2, y_2, z_2) \) in three-dimensional space is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of \( P(3, 4, 5) \) and \( A(0, 4, 0) \):
- \( x_1 = 3, y_1 = 4, z_1 = 5 \)
- \( x_2 = 0, y_2 = 4, z_2 = 0 \)

### Step 4: Calculate the differences
Now, calculate the differences:
- \( x_2 - x_1 = 0 - 3 = -3 \)
- \( y_2 - y_1 = 4 - 4 = 0 \)
- \( z_2 - z_1 = 0 - 5 = -5 \)

### Step 5: Substitute the differences into the distance formula
Now substitute these values into the distance formula:
\[ d = \sqrt{(-3)^2 + (0)^2 + (-5)^2} \]

### Step 6: Simplify the expression
Calculating the squares:
- \( (-3)^2 = 9 \)
- \( (0)^2 = 0 \)
- \( (-5)^2 = 25 \)

So, we have:
\[ d = \sqrt{9 + 0 + 25} = \sqrt{34} \]

### Conclusion
Thus, the perpendicular distance of the point \( (3, 4, 5) \) from the Y-axis is \( \sqrt{34} \).

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