Distance of the points (a,b,c) for the y axis is
(a) `sqrt(b^(2)+c^(2)) (b) sqrt(c^(2)+a^(2)) (c )sqrt(a^(2)+b^(2)) (d) sqrt(a^(2)+b^(2)+c^(2))`

Answer

To find the distance of the point \( (a, b, c) \) from the y-axis in three-dimensional space, we can follow these steps:

### Step 1: Understand the Position of the Point
The point \( (a, b, c) \) is represented in a 3D coordinate system where:
- \( a \) is the x-coordinate,
- \( b \) is the y-coordinate,
- \( c \) is the z-coordinate.

### Step 2: Identify the Y-Axis
In 3D space, the y-axis is represented by points of the form \( (0, y, 0) \). This means that the x-coordinate and z-coordinate are both zero.

### Step 3: Choose a Point on the Y-Axis
Let’s choose a point on the y-axis, which can be represented as \( (0, b, 0) \). This point has the same y-coordinate as our point \( (a, b, c) \).

### Step 4: Use the Distance Formula
The distance \( D \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in 3D space is given by the formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 5: Substitute the Coordinates
Substituting the coordinates of our points into the distance formula:
- Point 1: \( (a, b, c) \)
- Point 2: \( (0, b, 0) \)

We have:
\[ D = \sqrt{(0 - a)^2 + (b - b)^2 + (0 - c)^2} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ D = \sqrt{(-a)^2 + 0^2 + (-c)^2} = \sqrt{a^2 + c^2} \]

### Step 7: Conclusion
Thus, the distance of the point \( (a, b, c) \) from the y-axis is:
\[ D = \sqrt{a^2 + c^2} \]

### Step 8: Identify the Correct Option
Looking at the options provided:
- (a) \( \sqrt{b^2 + c^2} \)
- (b) \( \sqrt{c^2 + a^2} \)
- (c) \( \sqrt{a^2 + b^2} \)
- (d) \( \sqrt{a^2 + b^2 + c^2} \)

The correct answer is option (b) \( \sqrt{c^2 + a^2} \).