Problem 85 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A variable plane passes through a fixed point \((3,2,1)\) and meets \(x, y\) and \(z\) axes at \(A, B\) and \(C\) respectively. A plane is drawn parallel to \(y z-p\) lane through \(A\), a second plane is drawn parallel \(z x-\) plane through \(B\) and a third plane is drawn parallel to \(x y-\) plane through \(C\). Then the locus of the point of intersection of these three planes, is [Online April 15, 2018] (a) \(x+y+z=6\) (b) \(\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1\) (c) \(\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1\) (d) \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}\)

Step-by-Step Solution

Answer

(c) \(\frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1\).

Step 1: Equation of the plane

Let the equation of the plane be \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). Since the plane passes through the fixed point \( (3, 2, 1) \), substituting these coordinates gives \( \frac{3}{a} + \frac{2}{b} + \frac{1}{c} = 1 \).

Step 2: Intersection with axes

The plane intersects the x-axis at \((a,0,0)\), the y-axis at \((0,b,0)\), and the z-axis at \((0,0,c)\).

Step 3: Equation of planes parallel to coordinate planes

Plane through A parallel to yz-plane: \(x = a\), through B parallel to zx-plane: \(y = b\), and through C parallel to xy-plane: \(z = c\).

Step 4: Find the intersection of three planes

The intersection of these three planes is the point \((a, b, c)\).

Step 5: Substitute and find the locus

The point \((a, b, c)\) also satisfies \(\frac{3}{a} + \frac{2}{b} + \frac{1}{c} = 1\). Therefore, the locus of the point of intersection \((x, y, z)\) is also given by \(\frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that uses a coordinate system to define and examine the position of points and lines in space. It helps in analyzing geometric shapes and their properties using algebraic equations.

Understanding coordinate geometry is crucial because it allows us to visually grasp the relationships between various geometric elements. In our exercise, we dealt with a plane, a two-dimensional flat surface extending infinitely in three-dimensional space. This plane interacts with the coordinate axes at specific intercepts (points where it crosses the axes).

The equation we started with was \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), representing a plane in 3D space. By substituting the coordinates of the fixed point (3, 2, 1) through which the plane passes, we developed a specific relationship:

• Find the intersection points on the axes, located at \( (a,0,0) \), \( (0,b,0) \), and \( (0,0,c) \).
• This step is essential in visualizing the plane's position in the 3D coordinate system.

Thus, coordinate geometry provides us with a powerful way to capture and solve the problem of knowing exactly where and how the plane intersects and aligns itself within a 3D space.

Planes in 3D

When dealing with planes in three-dimensional space, we're interested in two main components: positioning and orientation. A plane is defined by its equation, which, in this context, contains variables representing the axes: x, y, and z.

Each plane forms a flat, two-dimensional surface. In our exercise, the plane meets each of the x, y, and z axes at points A, B, and C, respectively. Understanding these intersections helps determine the plane's spatial orientation.

Key factors about planes in 3D include:

• Each plane can be parallel to one of the coordinate planes (yz-plane, zx-plane, or xy-plane).
• The plane's equation aligns with a fixed point or line within 3D space to define its unique position.
• By modifying its intercepts (a, b, c), the plane shifts within this space, maintaining parallelism to specific axes or planes.

By using equations like \( x = a \), \( y = b \), and \( z = c \), planes parallel to the yz-, zx-, and xy-planes can be drawn to illustrate this concept further. Such representation in coordinate geometry allows for intuitive comprehension of 3D relationships and intersections.

Intersection of Planes

In coordinate geometry, the intersection of planes is a concept where multiple planes meet at a common point in a space defined by axes. The skill of finding the intersection point is critical in understanding and solving spatial problems within the 3D coordinate system.

Here's how it works in our exercise:

• We look at planes drawn parallel to each axis-aligned plane through the points A, B, and C.
• The point of intersection for these planes was deduced to be \( (a, b, c) \).
• To verify this, we use the original condition \( \frac{3}{a} + \frac{2}{b} + \frac{1}{c} = 1 \).

This equation ensures that the intersection point aligns with the given planes' parameters. The intersection is the key to finding the locus of a point, which is a path or set of points satisfying specific conditions in geometry. Solving for the locus tells us where the intersection point can travel under these constraints, similarly shown by the exercise equation \( \frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1 \). As a result, understanding plane intersections bridges geometry and algebra for solving complex spatial puzzles.