Problem 52 - A Complete Resource Book in Mathematics for JEE Main

Question

The intersection of the spheres \(x^{2}+y^{2}+z^{2}+7 x-2 y\) \(-z=13\) and \(x^{2}+y^{2}+z^{2}-3 x+3 y+4 z=8\) is the same as the intersection of one of the sphere and the plane (A) \(x-y-z=1\) (B) \(x-2 y-z=1\) (C) \(x--2 z=1\) (D) \(2 x-y-z=1\)

Step-by-Step Solution

Answer

The intersection is with plane (D): \(2x - y - z = 1\).

Step 1: Identify the Equations of the Spheres

Start with the given equations of the spheres. The first sphere is: \[ x^2 + y^2 + z^2 + 7x - 2y - z = 13 \] The second sphere is: \[ x^2 + y^2 + z^2 - 3x + 3y + 4z = 8. \] These expressions represent the equations of two spheres in a three-dimensional space.

Step 2: Eliminate Like Terms and Rearrange

Subtract the equation of the second sphere from the equation of the first sphere to eliminate the common terms \(x^2 + y^2 + z^2\). That results in:\[ (7x - 2y - z) - (-3x + 3y + 4z) = 13 - 8 \]Simplify to get: \[ 10x - 5y - 5z = 5. \]

Step 3: Simplify to Find the Plane

Divide the entire equation by 5 to simplify and express it in standard form. That gives:\[ 2x - y - z = 1. \]This is the equation of the plane that represents the intersection of the two spheres.

Step 4: Match with Given Options

Compare the equation of the plane found with the given options. The plane \(2x - y - z = 1\) exactly matches option (D).

Key Concepts

Spheres Intersection

When dealing with three-dimensional geometry, understanding the intersection of spheres is an intriguing concept. In this exercise, we start with two sphere equations in a three-dimensional coordinate system:

• First Sphere: \( x^2 + y^2 + z^2 + 7x - 2y - z = 13 \)
• Second Sphere: \( x^2 + y^2 + z^2 - 3x + 3y + 4z = 8 \)

Though sphere equations often appear complex, they represent a symmetrical shape in 3D space. An intersection between two spheres can occur in various ways: as a circle, a single point, or a null set if they don't intersect at all. In this scenario, subtracting one sphere's equation from the other simplifies to the equation of a plane. This plane represents the line of intersection common to both spheres. By focusing on like terms and eliminating them, we simplify the complexities of intersecting entities into a more manageable form.

Equation of a Plane

The resulting equation from the spheres' intersection often reduces to a plane equation. In this case, the equation after simplification is \[ 2x - y - z = 1. \] This form represents a standard plane equation in three-dimensional space and embodies a flat, two-dimensional surface.
A plane is essentially defined by the equation \( ax + by + cz = d \), where \(a\), \(b\), and \(c\) are coefficients determining the plane's orientation and \(d\) determines its distance from the origin.
Solving for the intersection boils down to simplifying the equation that emerges when working through the sphere equations. By eliminating similar terms, we find a unified form, directly linking back to the nature of planes in simple coordinate geometry.

Coordinate Geometry Concepts

To fully understand the concepts of sphere intersections and plane equations, one should be familiar with the basics of coordinate geometry.
In this context:

• Points: Defined by coordinates \((x, y, z)\) in a 3D system.
• Lines: Equations like \( x = a, y = b, z = c \) describe lines using parametric equations connecting two points.
• Planes: Represented with equations \( ax + by + cz = d \), offering a flat surface picture in three dimensions.

Coordinate geometry serves as the backbone for various geometric interpretations. By mapping geometrical figures like spheres and planes into equations, we solve many complex problems involving shapes and distances in space. These solutions often rely on the careful manipulation of algebraic expressions. The exercise shows that seemingly complex interactions can be well understood by applying systematic algebraic techniques to dissect and simplify overlapping geometric entities. Understanding how to transform these 3D figures into equations is essential in higher-level geometry.