If the perpendicular distance of the point `(6, 5, 8)` from the Y-axis is `5lambda` units, then `lambda` is equal to

Answer

To find the value of \( \lambda \) given that the perpendicular distance of the point \( (6, 5, 8) \) from the Y-axis is \( 5\lambda \) units, we can follow these steps:

### Step 1: Understand the Perpendicular Distance from the Y-axis
The Y-axis is defined by the points where \( x = 0 \) and \( z = 0 \). Therefore, the perpendicular distance from any point \( (x_1, y_1, z_1) \) to the Y-axis can be calculated using the formula:
\[ \text{Distance} = \sqrt{(x_1 - 0)^2 + (z_1 - 0)^2} \]

### Step 2: Substitute the Coordinates of the Given Point
For the point \( (6, 5, 8) \), we have:
- \( x_1 = 6 \)
- \( y_1 = 5 \) (not needed for distance calculation)
- \( z_1 = 8 \)

Substituting these values into the distance formula:
\[ \text{Distance} = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} \]

### Step 3: Calculate the Squares
Calculating the squares:
\[ 6^2 = 36 \]
\[ 8^2 = 64 \]

### Step 4: Add the Squares
Now, add the squares:
\[ 36 + 64 = 100 \]

### Step 5: Take the Square Root
Taking the square root gives us the distance:
\[ \text{Distance} = \sqrt{100} = 10 \]

### Step 6: Set the Distance Equal to \( 5\lambda \)
According to the problem, this distance is also equal to \( 5\lambda \):
\[ 10 = 5\lambda \]

### Step 7: Solve for \( \lambda \)
Now, solve for \( \lambda \):
\[ \lambda = \frac{10}{5} = 2 \]

Thus, the value of \( \lambda \) is \( 2 \).

### Final Answer
\[ \lambda = 2 \]

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