Problem 106 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the distance between planes, \(4 x-2 y-4 z+1=0\) and \(4 x-2 y-4 z+d=0\) is 7 , then dis: [Online April 12, 2014] (a) 41 or \(-42\) (b) 42 or \(-43\) (c) \(-41\) or 43 (d) \(-42\) or 44

Step-by-Step Solution

Answer

(c)

Step 1: Understanding the Distance Formula for Parallel Planes

The distance between two parallel planes given by their equations \( ax + by + cz + d_1 = 0 \) and \( ax + by + cz + d_2 = 0 \) is calculated using the formula: \( \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \).

Step 2: Substitute Known Values Into the Distance Formula

Since the normal vector coefficients \( a, b, c \) for both planes are identical \( 4, -2, -4 \) respectively, and the distance is given as 7, we can set up the equation: \[ \frac{|1 - d|}{\sqrt{4^2 + (-2)^2 + (-4)^2}} = 7 \]

Step 3: Calculate the Magnitude of the Normal Vector

Compute \( \sqrt{4^2 + (-2)^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \).

Step 4: Simplify the Equation

Substitute 6 from Step 3 into the distance formula equation: \[ \frac{|1 - d|}{6} = 7 \]. This simplifies to \( |1 - d| = 42 \).

Step 5: Solve for \( d \)

The absolute value equation \( |1 - d| = 42 \) leads to two possible scenarios: \( 1 - d = 42 \) or \( 1 - d = -42 \). Solve each case separately: 1. For \( 1 - d = 42 \), then \( d = 1 - 42 = -41 \).2. For \( 1 - d = -42 \), then \( d = 1 + 42 = 43 \).

Step 6: Verify Solutions Against the Choices

According to the problem's multiple-choice options, the solutions \( d = -41 \) or \( d = 43 \) match the answer choice (c).

Key Concepts

Distance Formula

The distance formula is a mathematical tool used to calculate the distance between two points or, in this case, between two planes. In the context of parallel planes, the formula becomes particularly useful as it helps us to quantify how far apart the planes are from each other. For planes represented by their equations, the distance formula specifically used is: \[ \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \] Where \(a\), \(b\), and \(c\) are the coefficients of \(x\), \(y\), and \(z\) in the plane equation, and \(d_1\) and \(d_2\) are the constant terms from each plane’s equation. This formula measures the perpendicular distance, ensuring that even if planes are staggered, the shortest distance is calculated. The numerator \(|d_1 - d_2|\) gives the absolute difference between the constant terms, which corresponds to how much one plane is shifted from the other along the direction of the normal vector. Simplifying the denominator \(\sqrt{a^2 + b^2 + c^2}\) provides the magnitude of the normal vector, which scales the distance appropriately according to the orientation of the planes.

Parallel Planes

Parallel planes are two or more planes in three-dimensional space that are always equidistant from each other. They never intersect or meet, regardless of how far you extend them in space. For planes to be parallel, they must have identical normal vectors. This is because the normal vector determines the orientation of the plane in space. If the orientations are the same, the planes remain parallel.

• Identical normal vectors: If the normal vectors of the planes are proportional (i.e., one is a scalar multiple of the other), the planes are parallel.
• Example of parallel plane equations: Consider \(4x - 2y - 4z + 1 = 0\) and \(4x - 2y - 4z + d = 0\). Here, both planes have the same normal vector \([4, -2, -4]\), indicating they are parallel.

Understanding parallel planes helps in determining spatial arrangements and calculating distances between different geometric objects in mathematics and real-world applications, like architectural planning or physics.

Plane Equations

Plane equations are mathematical representations of flat surfaces in three-dimensional space. They describe the orientation and position of a plane using a linear equation of the form: \[ ax + by + cz + d = 0 \] In this equation:

• \(a, b,\) and \(c\) are the coefficients that correspond to the direction or orientation of the plane.
• \(x, y,\) and \(z\) are the variables representing positional coordinates in 3D space.
• \(d\) is the constant term, affecting the plane's position relative to the origin.

These equations provide a foundation for describing and solving complex problems involving planes, such as finding the distance between planes or determining intersections with other geometric entities. In our exercise, the equation \(4x - 2y - 4z + 1 = 0\) describes a specific plane, and by changing \(d\) to another constant, we tweak the plane’s position without altering its orientation, hence managing its distance from the other plane.

Normal Vector

A normal vector is a critical concept when dealing with 3D plane geometry. It is a vector that is perpendicular to a given plane or surface. In the plane equation \(ax + by + cz + d = 0\), the vector \([a, b, c]\) acts as the normal vector. The normal vector is vital because:

• It provides information about the orientation of the plane. The direction in which the normal vector points tells you which way the plane is facing.
• It helps determine parallelism with other planes. If two planes have proportional normal vectors, they are parallel.
• Its magnitude \(\sqrt{a^2 + b^2 + c^2}\) is used in the distance formula for calculating the distance between parallel planes.

In our example, the normal vector \([4, -2, -4]\) denotes the orientation for both given planes \(4x - 2y - 4z + 1 = 0\) and \(4x - 2y - 4z + d = 0\). This common normal vector proves they are parallel and aids in computing the distance between them effectively.