Coordinate Geometry

We are given the points and , and the plane divides the line segment joining and in the ratio .
Step 1: Use the section formula
The section formula gives the coordinates of the point dividing the line segment in a given ratio.
If a point divides the line joining and in the ratio , the coordinates of the dividing point are:
Step 2: Apply the formula
In this case, the point divides the line joining and in the ratio .
We are working in the plane , so the -coordinates of both points are irrelevant to the solution. The coordinates of the dividing point are: Since the dividing point lies in the plane , the -coordinate must be zero: Solving , we get: Thus, the value of is , corresponding to option (b).
Step 3: Conclusion
The value of is , corresponding to option (b).

Step 1: Understanding the Geometry of the Points
The points are .
To check if they form a parallelogram, we check if the diagonals bisect each other.
The midpoint of the diagonal joining and is: The midpoint of the diagonal joining and is: These midpoints are the same, so the points form a parallelogram.
Step 2: Conclusion
Thus, the points form the vertical of a parallelogram.