Final answer:
To find the equation of the plane in the question, one must find the point of intersection of the given lines, then choose the plane farthest from the origin using the distance formula for a plane.
Step-by-step explanation:
The question is asking to determine the equation of a plane that passes through the point of intersection of two lines and has the largest distance from the origin. To find this plane, we must first find the point of intersection of the given lines, then use this point to construct the plane.
The first step is to equate the respective components of the two lines to find a common point of intersection that satisfies both lines:
- ((x - 1)/3) = ((y - 2)/1) = ((z - 3)/2)
- ((x - 3)/1) = ((y - 1)/2) = ((z - 2)/3)
Resolve these equations to find the coordinates x, y, and z for the intersection point. Once we have the point of intersection, we observe the given plane options (A to D) and determine which of them contains this point. Next, to find which plane is farthest from the origin, the distance formula for a plane, D = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2), can be applied where (x0, y0, z0) is the point on the plane, and A, B, C are the coefficients of the plane equation Ax + By + Cz + D = 0.
The plane with the largest numerator and the smallest denominator in the distance formula will be the furthest from the origin. By comparing these values for all given planes, we can find the plane that satisfies the condition. However, the necessary calculations and intermediate steps have not been provided in the question's context, so we cannot definitively choose one of the options (A to D).