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

Options

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

Explanation

To find the y-coordinate of point P, which lies on the line segment joining the points (3, 2, -1) and (6, 2, -2), and given that the x-coordinate of P is 5, we can follow these steps:

### Step 1: Understand the Problem
We have two points A(3, 2, -1) and B(6, 2, -2). Point P divides the line segment AB in some ratio. We need to find the y-coordinate of P when the x-coordinate is given as 5.

### Step 2: Set Up the Ratio
Let the ratio in which point P divides the line segment AB be λ:1. The coordinates of point P can be expressed using the section formula:
\[ P(x, y, z) = \left( \frac{x_1 + \lambda x_2}{\lambda + 1}, \frac{y_1 + \lambda y_2}{\lambda + 1}, \frac{z_1 + \lambda z_2}{\lambda + 1} \right) \]
where \( (x_1, y_1, z_1) = (3, 2, -1) \) and \( (x_2, y_2, z_2) = (6, 2, -2) \).

### Step 3: Find the x-coordinate
Since we know the x-coordinate of P is 5, we can set up the equation:
\[ 5 = \frac{3 + 6\lambda}{\lambda + 1} \]

### Step 4: Solve for λ
Cross-multiplying gives:
\[ 5(\lambda + 1) = 3 + 6\lambda \]
Expanding this, we have:
\[ 5\lambda + 5 = 3 + 6\lambda \]
Rearranging the terms:
\[ 5 - 3 = 6\lambda - 5\lambda \]
This simplifies to:
\[ 2 = \lambda \]

### Step 5: Substitute λ to Find the y-coordinate
Now that we have λ = 2, we can find the y-coordinate of point P using the y-coordinate formula:
\[ y = \frac{y_1 + \lambda y_2}{\lambda + 1} \]
Substituting the values:
\[ y = \frac{2 + 2 \cdot 2}{2 + 1} = \frac{2 + 4}{3} = \frac{6}{3} = 2 \]

### Final Answer
Thus, the y-coordinate of point P is **2**.

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