The shortest distance of the point (1, 3, 5) from
`x^(2) + y^(2) = 0` is

Options

A. 5 units
B. `sqrt(10) units
C. `2sqrt(5) units
D. `5/sqrt(2)` units

Answer

To find the shortest distance from the point \( (1, 3, 5) \) to the surface defined by the equation \( x^2 + y^2 = 0 \), we can follow these steps:

### Step 1: Understand the equation \( x^2 + y^2 = 0 \)
The equation \( x^2 + y^2 = 0 \) implies that both \( x \) and \( y \) must be zero. This is because the sum of two squares can only be zero if both squares are zero. Therefore, we have:
\[ x = 0 \quad \text{and} \quad y = 0 \]
This means that the only point that satisfies this equation in three-dimensional space is \( (0, 0, z) \) for any value of \( z \).

### Step 2: Identify the point of interest
The point we are interested in is \( P(1, 3, 5) \). We need to find the shortest distance from this point to the line defined by \( x = 0 \) and \( y = 0 \) (the z-axis).

### Step 3: Use the distance formula
The distance \( d \) from a point \( (x_1, y_1, z_1) \) to the z-axis (where \( x = 0 \) and \( y = 0 \)) can be calculated using the formula:
\[ d = \sqrt{(x_1 - 0)^2 + (y_1 - 0)^2} \]
Substituting the coordinates of point \( P(1, 3, 5) \):
\[ d = \sqrt{(1 - 0)^2 + (3 - 0)^2} = \sqrt{1^2 + 3^2} \]

### Step 4: Calculate the distance
Now, compute the values:
\[ d = \sqrt{1 + 9} = \sqrt{10} \]

### Step 5: Conclusion
The shortest distance from the point \( (1, 3, 5) \) to the z-axis is:
\[ \sqrt{10} \text{ units} \]