Plane

Concept: 3D Geometry - Equation of a Plane and Cross Product of Normal Vectors.

Step 1: Set up the general equation for the required plane. The equation of any plane passing through a given point can be written as , where are the direction ratios of the plane's normal. For the point , the equation is .

Step 2: Establish the first perpendicularity condition. If two planes are perpendicular, their normal vectors are orthogonal, meaning their dot product is zero ( ). Our required plane is perpendicular to . The normal vector of this plane is . This gives our first condition: .

Step 3: Establish the second perpendicularity condition. Similarly, our required plane is perpendicular to the second plane , whose normal vector is . This gives our second condition: .

Step 4: Solve for the direction ratios using cross-multiplication. We have a system of two homogeneous linear equations: 1) 2) We solve for proportional values of using cross-multiplication: . We can assign these proportional values directly to the direction ratios: , , .

Step 5: Substitute the direction ratios back to form the final equation. Substitute , , and back into the general equation from
Step 1: . Expand the terms: . Combine the constant terms: . This matches option A. $ $

The given equation for the pair of straight lines is: One of the lines is . Substituting into the equation: Simplifying the terms: Multiplying through by 16 to eliminate the fractions: Solving for : Final Answer: