P is a point on the line segment joining the points `(3,2,-1) and (6,2,-2)`. If the x-coordinate of P is `5`, then its y-coordinate is

Options

A. `1`
B. `-1`
C. `2`
D. `-2`

Answer

To find the y-coordinate of point P on the line segment joining the points (3, 2, -1) and (6, 2, -2) where the x-coordinate of P is 5, we can use the section formula. Here’s a step-by-step solution:

### Step 1: Identify the points and the given x-coordinate
Let \( A(3, 2, -1) \) and \( B(6, 2, -2) \) be the endpoints of the line segment. The coordinates of point P are \( (5, y, z) \) where we need to find the value of \( y \).

### Step 2: Use the section formula
The section formula states that if a point P divides the line segment joining points A and B in the ratio \( k:1 \), then the coordinates of P can be expressed as:
\[ P\left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}, \frac{kz_2 + z_1}{k + 1} \right) \]
where \( (x_1, y_1, z_1) \) are the coordinates of point A and \( (x_2, y_2, z_2) \) are the coordinates of point B.

### Step 3: Set up the equation for the x-coordinate
Given that the x-coordinate of P is 5, we can set up the equation:
\[ 5 = \frac{k \cdot 6 + 3}{k + 1} \]

### Step 4: Solve for k
Multiply both sides by \( k + 1 \):
\[ 5(k + 1) = 6k + 3 \]
Expanding gives:
\[ 5k + 5 = 6k + 3 \]
Rearranging the equation:
\[ 5k + 5 - 3 = 6k \]
\[ 2 = k \]

### Step 5: Find the y-coordinate using k
Now that we have \( k = 2 \), we can find the y-coordinate using the section formula:
\[ y = \frac{ky_2 + y_1}{k + 1} = \frac{2 \cdot 2 + 2}{2 + 1} \]
Calculating this gives:
\[ y = \frac{4 + 2}{3} = \frac{6}{3} = 2 \]

### Conclusion
Thus, the y-coordinate of point P is \( 2 \).