Question

If \( Q(1, 0, 1) \) is the image of the point \( P(a, b, c) \) in the line \( \frac{x + 1}{2} = \frac{y - 3}{-2} = \frac{z}{-1} \), then \( a + b + c \) is equal to:

• A. 4
• B. 2
• C. 0
• D. -2

Hint:

In reflection problems, the midpoint of the original point and its image lies on the given line or plane.

Answer 1

The Correct Option is C

Step 1: Write the parametric equations for the line.
The given line equation is \( \frac{x + 1}{2} = \frac{y - 3}{-2} = \frac{z}{-1} \). Let the common parameter be \( t \). The parametric equations for the line are:
\[ x = 2t - 1, \quad y = -2t + 3, \quad z = -t \]

Step 2: Write the formula for reflection.
The reflection formula for a point \( P(a, b, c) \) and its image \( Q(1, 0, 1) \) gives the midpoint between \( P \) and \( Q \). The midpoint lies on the line.

Step 3: Calculate the midpoint.
The midpoint \( M \) is
\[ M = \left( \frac{a + 1}{2}, \frac{b + 0}{2}, \frac{c + 1}{2} \right) \]

Step 4: The midpoint lies on the line.
Substitute the midpoint \( M \) into the parametric equations of the line:
\[ \frac{a + 1}{2} = \frac{b + 0}{-2} = \frac{c + 1}{-1} \]

Step 5: Solve the system of equations.
From \( \frac{a + 1}{2} = \frac{b}{-2} \), we get \( a = -b - 1 \). From \( \frac{a + 1}{2} = \frac{c + 1}{-1} \), we get \( a = -2c - 3 \).

Step 6: Solve for \( a + b + c \).
From \( a = -b - 1 \) and \( a = -2c - 3 \), we find \( b = 2c + 2 \)
and \( a = -2c - 3 \).
Thus, \( a + b + c = 0 \).