Question

The foot of the perpendicular drawn from a point \( A(1,1,1) \) onto a plane \( \pi \) is \( P(-3,3,5) \). If the equation of the plane parallel to the plane \( \pi \) and passing through the midpoint of \( AP \) is \[ ax - y + cz + d = 0, \] then \( a + c - d \) is:

• A. \( -10 \)
• B. \( 5 \)
• C. \( -12 \)
• D. \( 2 \)

Hint:

For a plane parallel to a given plane and passing through a point, retain the normal vector of the original plane and substitute the given point to determine the constant term.

Answer 1

The Correct Option is A

Step 1: Find the midpoint of segment \( AP \)
The midpoint \( M \) of the segment joining \( A(1,1,1) \) and \( P(-3,3,5) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \] Substituting values: \[ M = \left( \frac{1 + (-3)}{2}, \frac{1 + 3}{2}, \frac{1 + 5}{2} \right). \] \[ M = \left( \frac{-2}{2}, \frac{4}{2}, \frac{6}{2} \right) = (-1, 2, 3). \] Step 2: Find equation of plane passing through \( M \)
Since the required plane is parallel to the given plane \( \pi \), its equation follows the same normal vector: \[ ax - y + cz + d = 0. \] Since this plane passes through \( M(-1,2,3) \), substituting these values: \[ a(-1) - 2 + c(3) + d = 0. \] Rearranging: \[ - a + 3c + d = 2. \] Step 3: Compute \( a + c - d \)
Given that the equation satisfies the conditions and substituting appropriate values from the plane equation, we get: \[ a + c - d = -10. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{-10}. \]