What is the number of possible values of k for which the line joining the points (k,1,3) and (1,-2,k+1) also passes through the point (15,2,-4)?

Options

A. Zero
B. One
C. Two
D. Infinite

Answer

To find the number of possible values of \( k \) for which the line joining the points \( (k, 1, 3) \) and \( (1, -2, k+1) \) also passes through the point \( (15, 2, -4) \), we can follow these steps:

### Step 1: Write the parametric equations of the line
The line joining the two points can be expressed in terms of a parameter \( t \). The parametric equations for the line can be derived from the coordinates of the two points:

\[ x = k + t(1 - k) \]
\[ y = 1 + t(-2 - 1) = 1 - 3t \]
\[ z = 3 + t((k + 1) - 3) = 3 + t(k - 2) \]

### Step 2: Substitute the point (15, 2, -4) into the parametric equations
We need to find the value of \( t \) for which the point \( (15, 2, -4) \) lies on the line. Therefore, we set up the equations:

1. \( 15 = k + t(1 - k) \)
2. \( 2 = 1 - 3t \)
3. \( -4 = 3 + t(k - 2) \)

### Step 3: Solve for \( t \) from the second equation
From the second equation:

\[ 2 = 1 - 3t \implies 3t = 1 - 2 \implies 3t = -1 \implies t = -\frac{1}{3} \]

### Step 4: Substitute \( t \) into the first equation
Now substitute \( t = -\frac{1}{3} \) into the first equation:

\[ 15 = k + \left(-\frac{1}{3}\right)(1 - k) \]
This simplifies to:

\[ 15 = k - \frac{1 - k}{3} \]
Multiplying through by 3 to eliminate the fraction:

\[ 45 = 3k - (1 - k) \]
\[ 45 = 3k - 1 + k \]
\[ 45 + 1 = 4k \implies 46 = 4k \implies k = \frac{46}{4} = 11.5 \]

### Step 5: Substitute \( t \) into the third equation
Now substitute \( t = -\frac{1}{3} \) into the third equation:

\[ -4 = 3 + \left(-\frac{1}{3}\right)(k - 2) \]
This simplifies to:

\[ -4 = 3 - \frac{k - 2}{3} \]
Multiplying through by 3:

\[ -12 = 9 - (k - 2) \]
\[ -12 = 9 - k + 2 \]
\[ -12 = 11 - k \implies k = 11 + 12 = 23 \]

### Step 6: Determine the possible values of \( k \)
From the two equations, we found \( k = 11.5 \) and \( k = 23 \). Therefore, there are two possible values of \( k \).

### Final Answer
The number of possible values of \( k \) is **2**.