Problem 17 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The plane \(x+y+z=5 \sqrt{3}\) and sphere \(x^{2}+y^{2}+z^{2}=5\) (A) touch each other (B) cut in a circle (C) do not meet (D) none of these

Step-by-Step Solution

Answer

The sphere and the plane do not meet.

Step 1: Identify the Problem

We need to determine the relationship between the plane and the sphere given by the equations \(x+y+z=5 \sqrt{3}\) and \(x^2 + y^2 + z^2 = 5\). The options are whether they touch each other, cut in a circle, or do not meet.

Step 2: Find the Center and Radius of the Sphere

The equation of the sphere \(x^2 + y^2 + z^2 = 5\) represents a sphere centered at the origin \((0, 0, 0)\) with radius \(\sqrt{5}\).

Step 3: Find the Perpendicular Distance from Center to the Plane

The distance \(d\) from the point \((0, 0, 0)\) to the plane \(x+y+z=5\sqrt{3}\) is given by \(d = \frac{|0+0+0-5\sqrt{3}|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{5\sqrt{3}}{\sqrt{3}} = 5\).

Step 4: Compare the Distance with the Sphere's Radius

The perpendicular distance from the center of the sphere to the plane is 5, which is equal to the radius of the sphere, \(\sqrt{5}\approx 2.236\). This implies that the plane and sphere are not intersecting and the plane is entirely outside the sphere, hence they don't meet.

Key Concepts

Sphere Equation

The sphere equation is a critical geometric concept in understanding the relationship between a sphere and another geometric entity, such as a plane. A sphere in three-dimensional space is defined by a standard equation of the form:
\[ x^2 + y^2 + z^2 = r^2 \]where

• - \( (x, y, z) \) represents any point on the surface of the sphere,
• - \( r \) is the radius of the sphere, a constant that indicates the distance from the center to any point on the surface.

In the given problem, the equation \( x^2 + y^2 + z^2 = 5 \) specifies a sphere centered at the origin \( (0, 0, 0) \) and having a radius \( \sqrt{5} \) (approximately 2.236).
To comprehend this better, visualize the sphere as a perfectly round 3D object where any point is exactly \( \sqrt{5} \) units from the origin. Understanding the equation helps in determining how the sphere might interact with other shapes, such as where they might touch or intersect.

Plane Equation

A plane equation provides a representation of a flat surface extending infinitely in three dimensions. It is typically expressed in the general form:
\[ ax + by + cz = d \]where

• - \( (x, y, z) \) is a coordinate on the plane,
• - \( a, b, c \) are the coefficients that define the orientation of the plane,
• - \( d \) is a constant that positions the plane relative to the origin.

In our case, the equation \( x + y + z = 5\sqrt{3} \) describes a specific plane. The coefficients (1, 1, 1) determine that it has equal weighting in each of the x, y, and z directions, which means it has a diagonal orientation in the 3D space.
Understanding the plane equation is crucial because it helps determine relationships with other geometric figures, such as how close it lies to a sphere or whether they cross at some point.
The constant \( 5\sqrt{3} \) influences its height above the origin, playing a key role in how it relates spatially to the sphere.

Sphere and Plane Intersection

The intersection of a sphere and a plane can be visualized in various ways, leading to different types of interactions. These interactions are categorized based on the spatial relations between the objects:

• - **Touching**: When the plane is tangent to the sphere, meaning they meet at exactly one point.
• - **Intersecting in a circle**: When part of the plane slices through the sphere, creating a circular cross-section.
• - **No intersection**: When the plane does not intersect the sphere at all and remains outside.

To determine the type of interaction, we use the distance between the center of the sphere and the plane and compare it with the sphere's radius.
For our exercise, the sphere with radius \( \sqrt{5} \) and the perpendicular distance from its center to the plane is calculated as 5. Since 5 is greater than \( \sqrt{5} \), the entire plane sits outside the sphere, indicating no intersection whatsoever.
This comprehensive understanding confirms that our sphere and plane do not meet or intersect, compelling the conclusion that they occupy separate spaces in the 3D realm.