In sub - part (i) and (ii) choose the correct option and in sub - part (iii) to (v) answer the questions as instructed.
The length of the perpendicular drawn from the point P(3,4,5) on y- axis is

Options

A. `+-sqrt(34)`
B. `sqrt(34)`
C. `+-sqrt(43)`
D. `sqrt(43)`

Answer

To find the length of the perpendicular drawn from the point \( P(3, 4, 5) \) to the y-axis, we can follow these steps:

### Step 1: Identify the coordinates of the point on the y-axis
The y-axis is defined by the points where the x-coordinate and z-coordinate are both zero. Therefore, any point on the y-axis can be represented as \( (0, y, 0) \).

### Step 2: Determine the coordinates of the foot of the perpendicular
Since we are drawing a perpendicular from point \( P(3, 4, 5) \) to the y-axis, the foot of the perpendicular, let's call it point \( Q \), will have coordinates \( (0, 4, 0) \). This is because the y-coordinate remains the same (4) while the x and z coordinates become zero.

### Step 3: Use the distance formula to find the length of the perpendicular
The distance \( PQ \) between points \( P(3, 4, 5) \) and \( Q(0, 4, 0) \) can be calculated using the distance formula in 3D space:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates of points \( P \) and \( Q \):

\[ d = \sqrt{(0 - 3)^2 + (4 - 4)^2 + (0 - 5)^2} \]

### Step 4: Simplify the expression
Calculating each term inside the square root:

\[ d = \sqrt{(-3)^2 + (0)^2 + (-5)^2} \]
\[ d = \sqrt{9 + 0 + 25} \]
\[ d = \sqrt{34} \]

### Step 5: Conclusion
Thus, the length of the perpendicular drawn from the point \( P(3, 4, 5) \) to the y-axis is \( \sqrt{34} \) units.

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