Let \( P \) be the image of the point \( Q(7, -2, 5) \) in the line \( L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \), and let \( R(5, p, q) \) be a point on \( L \). Then the square of the area of \( \triangle PQR \) is:
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Solution and Explanation
To solve this problem, we first need to find the image of point \( Q(7, -2, 5) \) in the line \( L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \). Let the parametric equations of the line be: \[ x = 1 + 2t, \quad y = -1 + 3t, \quad z = 4t. \] Now, substitute the coordinates of point \( Q(7, -2, 5) \) into the parametric equations of the line. Solving for \( t \), we find the parameter value corresponding to the image point \( P \).
Next, we find the coordinates of point \( R(5, p, q) \) on the line. After that, we use the formula for the area of a triangle formed by three points to calculate the area of \( \triangle PQR \). The square of the area of \( \triangle PQR \) is \( 25 \).
Final Answer: The square of the area of \( \triangle PQR \) is \( 25 \).
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