Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x - y + z + 3 = 0 and let R(3, 5, \(\gamma\)) be a point of this plane. Then the square of the length of the line segment SR is _________.
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This problem is a great example of how using geometric properties can simplify a problem. Calculating the coordinates of the mirror image S would be a much longer process. Recognizing that the distance from any point on a reflection plane to an object is the same as the distance to its image (\(QR=SR\)) is the key shortcut.
Step 1: Understand the property of a mirror image.
The definition of a mirror image S of a point Q with respect to a plane is that for any point R on the plane, the distance from Q to R is equal to the distance from S to R. That is, \(SR = QR\). Therefore, \(SR^2 = QR^2\). Our goal is to find \(QR^2\).
Step 2: Find the complete coordinates of point R.
We are given that the point R(3, 5, \(\gamma\)) lies on the plane \(2x - y + z + 3 = 0\). To find \(\gamma\), we substitute the coordinates of R into the plane's equation:
\[ 2(3) - (5) + \gamma + 3 = 0 \]
\[ 6 - 5 + \gamma + 3 = 0 \]
\[ 1 + \gamma + 3 = 0 \]
\[ \gamma + 4 = 0 \implies \gamma = -4 \]
So, the point R is (3, 5, -4).
Step 3: Calculate the square of the distance between Q and R.
The coordinates are Q(1, 3, 4) and R(3, 5, -4).
Using the distance formula in 3D, \(d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2\):
\[ QR^2 = (3 - 1)^2 + (5 - 3)^2 + (-4 - 4)^2 \]
\[ QR^2 = (2)^2 + (2)^2 + (-8)^2 \]
\[ QR^2 = 4 + 4 + 64 = 72 \]
Step 4: State the final answer.
Since \(SR^2 = QR^2\), the square of the length of the line segment SR is 72.
