The plane \(2x−y+z=4\) intersects the line segment joining the points \(A(a,−2,4)\) and \(B(2,b,−3)\) at the point C in the ratio \(2:1\) and the distance of the point C from the origin is \(\sqrt5\). If \(ab<0 \) and \(P\) is the point \((a−b,b,2b−a)\). Then \(CP^2 \) is equal to
- \(\frac{16}{3}\)
- \(\frac{17}{3}\)
- \(\frac{73}{3}\)
- \(\frac{97}{3}\)
The Correct Option is B
Solution and Explanation
(A)The coordinates of \( C \) using the section formula: \[ C = \left(\frac{a + 4}{3}, \frac{2b - 2}{3}, \frac{2 - 2}{3}\right). \] (B)Substituting \( C \) into the plane equation \( 2x - y + z = 4 \): \[ 2\left(\frac{a + 4}{3}\right) - \left(\frac{2b - 2}{3}\right) + \left(\frac{2}{3}\right) = 4. \] Simplify: \[ \frac{2a + 8 + 2b - 2 + 2}{3} = 4 \implies 2a + 2b = 4 \implies a + b = 2. \quad (1) \] (C)The distance of \( C \) from the origin: \[ \left(\frac{a + 4}{3}\right)^2 + \left(\frac{2b - 2}{3}\right)^2 + \left(\frac{2}{3}\right)^2 = 5. \] Solve using \( a + b = 2 \) and simplify: \[ (b + 6)^2 + (2b - 2)^2 = 41 \implies 5b^2 + 4b - 1 = 0. \] (D)Roots are \( b = -1 \) or \( b = \frac{1}{5} \). Using \( ab < 0 \), \( (a, b) = (1, -1) \). (E)Coordinates of \( C \): \[ C = \left(\frac{5}{3}, \frac{-4}{3}, \frac{2}{3}\right). \] Coordinates of \( P \): \[ P = (2, -1, -3). \] (F) Compute \( CP^2 \): \[ CP^2 = \left(\frac{5}{3} - 2\right)^2 + \left(\frac{-4}{3} + 1\right)^2 + \left(\frac{2}{3} + 3\right)^2. \] Simplify: \[ CP^2 = \frac{17}{3}. \]
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Concepts Used:
Coordinate Geometry
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.