Question

Let line L pass through the point \( (0, 1, 2) \), intersect the line \( {2 = {3 = {4 \), and be parallel to the plane \( 2x + y - 3z = 4 \). Find the distance of the point \( P(1, -9, 2) \) from line L. \\

• A. \(9\)
• B. \(\sqrt{54}\)
• C. \(\sqrt{69}\)
• D. \(\sqrt{74}\)

Hint:

\begin{tcolorbox}[colback=white, colframe=black, colbacktitle=white, coltitle=black, title=QuickTip, width=\textwidth, halign=left] For coplanarity problems, use the condition that the determinant of the matrix formed by the vectors must be zero. \end{tcolorbox} \vspace{0.5cm} \hline \vspace{0.5cm}

Answer 1

The Correct Option is B

\\\textbf{Step 1: Write the position vectors of the points.} \\Let the points be:\[A = (1, -2, 3), \quad B = (2, -3, 4), \quad C = (\alpha + 1, 0, 2), \quad D = (9, \alpha - 8, 6)\]\textbf{Step 2: Use the condition for coplanarity.} \\The condition for coplanarity of four points is given by the determinant of the matrix formed by the vectors:\[\begin{vmatrix} 1 & -1 & 1 \\ \alpha & 2 & -1 \\ 8 & \alpha - 6 & 3\end{vmatrix} = 0\]Expanding the determinant:\[(6 + \alpha - 6) + 1(3\alpha + 8) + (\alpha^2 - 6\alpha - 16) = 0\]Simplifying:\[\alpha^2 - 2\alpha - 8 = 0\]\textbf{Step 3: Solve for \( \alpha \).} \\Solving the quadratic equation:\[\alpha = 4, -2\]\textbf{Step 4: Find the sum of all values of \( \alpha \).} \\The sum of all values of \( \alpha \) is:\[4 + (-2) = 2\]\textbf{Final Answer:} The sum of all values of \( \alpha \) is \( 2 \).