The equation to the straight line passing through the points (4,-5,2) and (-1,5,3) is

Options

A. `(x-4)/(1)=(y+5)/(-2)=(z+2)/(-1)`
B. `(x+1)/(1)=(y-5)/(2)=(z-3)/(-1)`
C. `(x)/(-1)=(y)/(5)=(z)/(3)`
D. `(x)/(4)=(y)/(-5)=(z)/(-2)`

Explanation

To find the equation of the straight line passing through the points (4, -5, 2) and (-1, 5, 3), we can use the formula for the equation of a line in three-dimensional space. The formula is given by:

\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \]

### Step 1: Identify the points
Let the first point be \( P_1(4, -5, 2) \) and the second point be \( P_2(-1, 5, 3) \).
Here, \( x_1 = 4, y_1 = -5, z_1 = 2 \) and \( x_2 = -1, y_2 = 5, z_2 = 3 \).

### Step 2: Calculate the differences
Now, we calculate the differences:
- \( x_2 - x_1 = -1 - 4 = -5 \)
- \( y_2 - y_1 = 5 - (-5) = 5 + 5 = 10 \)
- \( z_2 - z_1 = 3 - 2 = 1 \)

### Step 3: Substitute into the formula
Now we substitute these values into the formula:

\[ \frac{x - 4}{-5} = \frac{y + 5}{10} = \frac{z - 2}{1} \]

### Step 4: Rearranging the equation
We can rearrange the equation to express it in a more standard form:

1. From \( \frac{x - 4}{-5} = \frac{y + 5}{10} \):
\[ 10(x - 4) = -5(y + 5) \]
Simplifying gives:
\[ 10x - 40 = -5y - 25 \implies 10x + 5y = 15 \implies 2x + y = 3 \]

2. From \( \frac{y + 5}{10} = \frac{z - 2}{1} \):
\[ z - 2 = \frac{1}{10}(y + 5) \implies z = \frac{1}{10}y + \frac{5}{10} + 2 = \frac{1}{10}y + 0.5 + 2 = \frac{1}{10}y + 2.5 \]

### Final Equation
Thus, the equations of the line in parametric form can be expressed as:
\[ \frac{x - 4}{-5} = \frac{y + 5}{10} = z - 2 \]

### Summary of the Solution
The equation of the straight line passing through the points (4, -5, 2) and (-1, 5, 3) is:
\[ \frac{x - 4}{-5} = \frac{y + 5}{10} = z - 2 \]