A line joining the points (1,1,1) and (2,2,2) intersect the plane x+y+z=9 at the point

Options

A. (3,4,2)
B. (2,3,4)
C. (3,2,4)
D. (3,3,3)

Answer

To find the point where the line joining the points (1, 1, 1) and (2, 2, 2) intersects the plane defined by the equation \(x + y + z = 9\), we can follow these steps:

### Step 1: Determine the parametric equations of the line
The line can be represented in parametric form using the two points (1, 1, 1) and (2, 2, 2). The direction ratios of the line can be found by subtracting the coordinates of the two points:

\[ \text{Direction ratios} = (2 - 1, 2 - 1, 2 - 1) = (1, 1, 1) \]

Using the point (1, 1, 1) as the starting point, we can write the parametric equations of the line as:

\[ x = 1 + t, \quad y = 1 + t, \quad z = 1 + t \]

where \(t\) is a parameter.

### Step 2: Substitute the parametric equations into the plane equation
The equation of the plane is given by:

\[ x + y + z = 9 \]

Now, substituting the parametric equations into the plane equation:

\[ (1 + t) + (1 + t) + (1 + t) = 9 \]

### Step 3: Simplify the equation
Simplifying the left side gives:

\[ 3 + 3t = 9 \]

### Step 4: Solve for the parameter \(t\)
Subtract 3 from both sides:

\[ 3t = 6 \]

Now, divide by 3:

\[ t = 2 \]

### Step 5: Find the coordinates of the intersection point
Now that we have \(t\), we can substitute it back into the parametric equations to find the coordinates of the intersection point:

\[ x = 1 + 2 = 3, \quad y = 1 + 2 = 3, \quad z = 1 + 2 = 3 \]

Thus, the intersection point is:

\[ (3, 3, 3) \]

### Conclusion
The line joining the points (1, 1, 1) and (2, 2, 2) intersects the plane \(x + y + z = 9\) at the point \((3, 3, 3)\).

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