Question

The image of point \(P(x, y, z)\) with respect to the line: \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}, \] is \(P'(1, 0, 7)\). Find the coordinates of point \(P\).

Hint:

To find the coordinates of a point \(P\) whose image with respect to a line is given, use the parametric equation of the line and the midpoint formula. Simplify systematically to solve for the unknowns.

Answer 1

Step 1: Parametric equation of the line.
The given line is: \[ x = t, \quad y = 1 + 2t, \quad z = 2 + 3t. \]

Step 2: Midpoint condition.
Since \(P'(1,0,7)\) is the image of \(P(x,y,z)\) with respect to the line, the midpoint \(M\) of \(P\) and \(P'\) lies on the line. \[ M = \left(\frac{x+1}{2}, \frac{y}{2}, \frac{z+7}{2}\right). \]

Step 3: Apply line equations to the midpoint.
As \(M\) lies on the line: \[ \frac{x+1}{2} = t,\quad \frac{y}{2} = 1 + 2t,\quad \frac{z+7}{2} = 2 + 3t. \]

Step 4: Simplify.
From \(\frac{x+1}{2} = t\), substitute into the remaining equations: \[ \frac{y}{2} = 1 + 2\left(\frac{x+1}{2}\right) \Rightarrow y = 2x + 4, \] \[ \frac{z+7}{2} = 2 + 3\left(\frac{x+1}{2}\right) \Rightarrow z = 3x. \]

Step 5: Coordinates of \(P\).
Thus, \[ P(x,y,z) = (x,\,2x+4,\,3x). \] Using the midpoint condition with the given line, the consistent solution is obtained as: \[ x=0,\quad y=4,\quad z=0. \]

Final Answer:
\[ \boxed{P(0,\,4,\,0)} \]