If the points (x,y,-3), (2,0,-1) and (4,2,3) lie on a Straight line, then what is the value of x and y respectively?

Options

A. `1,-1`
B. `-1,1`
C. `0,2`
D. `3,4`

Answer

To find the values of \( x \) and \( y \) such that the points \( (x, y, -3) \), \( (2, 0, -1) \), and \( (4, 2, 3) \) lie on a straight line, we can use the condition for collinearity of three points in 3D space.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be:
- \( P_1 = (x, y, -3) \)
- \( P_2 = (2, 0, -1) \)
- \( P_3 = (4, 2, 3) \)

2. **Use the Collinearity Condition**:
The points \( P_1 \), \( P_2 \), and \( P_3 \) are collinear if the following ratio holds:
\[ \frac{x_3 - x_2}{x_3 - x_1} = \frac{y_3 - y_2}{y_3 - y_1} = \frac{z_3 - z_2}{z_3 - z_1} \]
Here, we assign:
- \( (x_1, y_1, z_1) = (x, y, -3) \)
- \( (x_2, y_2, z_2) = (2, 0, -1) \)
- \( (x_3, y_3, z_3) = (4, 2, 3) \)

3. **Set Up the Ratios**:
We can write the ratios as:
\[ \frac{4 - 2}{4 - x} = \frac{2 - 0}{2 - y} = \frac{3 - (-1)}{3 - (-3)} \]

4. **Calculate Each Ratio**:
- For the first ratio:
\[ \frac{2}{4 - x} \]
- For the second ratio:
\[ \frac{2}{2 - y} \]
- For the third ratio:
\[ \frac{4}{6} = \frac{2}{3} \]

5. **Set Up Equations**:
Now we can set up the equations:
\[ \frac{2}{4 - x} = \frac{2}{3} \]
and
\[ \frac{2}{2 - y} = \frac{2}{3} \]

6. **Solve for \( x \)**:
From the first equation:
\[ 2 \cdot 3 = 2 \cdot (4 - x) \]
Simplifying gives:
\[ 6 = 8 - 2x \]
Rearranging:
\[ 2x = 8 - 6 \implies 2x = 2 \implies x = 1 \]

7. **Solve for \( y \)**:
From the second equation:
\[ 2 \cdot 3 = 2 \cdot (2 - y) \]
Simplifying gives:
\[ 6 = 4 - 2y \]
Rearranging:
\[ 2y = 4 - 6 \implies 2y = -2 \implies y = -1 \]

8. **Final Values**:
Thus, the values of \( x \) and \( y \) are:
\[ x = 1, \quad y = -1 \]

### Conclusion:
The values of \( x \) and \( y \) are \( 1 \) and \( -1 \) respectively.