A point R with x - coordinate 4 lies on the line segment joining the points P(2,-3,4) and Q(8, 0, 10) . The coordinates of R are

Options

A. (4, -2, 6)
B. (-4,-2,6)
C. (-4,2,6)
D. (4,2,6)

Answer

To find the coordinates of the point R that lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10) with the x-coordinate of R being 4, we will use the section formula. Here’s the step-by-step solution:

### Step 1: Identify the coordinates of points P and Q
The coordinates of point P are given as \( P(2, -3, 4) \) and the coordinates of point Q are given as \( Q(8, 0, 10) \).

### Step 2: Use the section formula
Let R divide the line segment PQ in the ratio \( k:1 \). According to the section formula, the coordinates of point R can be expressed as:
\[ R\left(\frac{8k + 2}{k + 1}, \frac{0k - 3}{k + 1}, \frac{10k + 4}{k + 1}\right) \]

### Step 3: Set the x-coordinate of R
Since we know the x-coordinate of R is 4, we can set up the equation:
\[ \frac{8k + 2}{k + 1} = 4 \]

### Step 4: Solve for k
Cross-multiply to solve for k:
\[ 8k + 2 = 4(k + 1) \]
Expanding the right side:
\[ 8k + 2 = 4k + 4 \]
Now, rearranging gives:
\[ 8k - 4k = 4 - 2 \]
\[ 4k = 2 \]
\[ k = \frac{1}{2} \]

### Step 5: Find the y and z coordinates of R
Now that we have \( k = \frac{1}{2} \), we can find the y and z coordinates of R using the section formula.

**For y-coordinate:**
\[ y = \frac{0 \cdot \frac{1}{2} - 3}{\frac{1}{2} + 1} = \frac{0 - 3}{\frac{3}{2}} = \frac{-3}{\frac{3}{2}} = -2 \]

**For z-coordinate:**
\[ z = \frac{10 \cdot \frac{1}{2} + 4}{\frac{1}{2} + 1} = \frac{5 + 4}{\frac{3}{2}} = \frac{9}{\frac{3}{2}} = 6 \]

### Step 6: Write the coordinates of R
Thus, the coordinates of point R are:
\[ R(4, -2, 6) \]

### Final Answer
The coordinates of R are \( (4, -2, 6) \).

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