Question

If the image of the point \( P(3, 2, a) \) reflected about the line \[ \frac{x-3}{2} = \frac{y-5}{5} = \frac{z-2}{-2} \] is \( (5, b, c) \), then the value of \( \sigma^2 + b^2 + c^2 \) is:

• A. \( \frac{4849}{8} \)
• B. \( \frac{4245}{4} \)
• C. \( \frac{3947}{8} \)
• D. \( \frac{2429}{4} \)

Hint:

When reflecting a point across a line, the coordinates of the point and the line's direction vector are used to find the new reflected point.

Answer 1

The Correct Option is A

Step 1: Equation of the line.
The given line equation can be written in parametric form: \[ x = 3 + 2t, \quad y = 5 + 5t, \quad z = 2 - 2t \] where \( t \) is the parameter. Step 2: Reflecting the point.
To reflect a point across a line, we first find the direction vector of the line and then use the formula for reflection of a point across a line. After finding the reflected coordinates \( (5, b, c) \), we can substitute these values into the expression for \( \sigma^2 + b^2 + c^2 \). Step 3: Conclusion.
Thus, \( \sigma^2 + b^2 + c^2 = \frac{4849}{8} \). Final Answer: \[ \boxed{\frac{4849}{8}} \]