Problem 30 - A Complete Resource Book in Mathematics for JEE Main

Question

The plane \(l x+m y=0\) is rotated about its line of intersection with the plane \(z=0\) through an angle \(\alpha\). The equation of the plane in its new position is (A) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \sin \alpha=0\) (B) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \tan \alpha=0\) (C) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \cot \alpha=0\) (D) none of these

Step-by-Step Solution

Answer

The correct equation of the plane in its new position is (A) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \sin \alpha=0\).

Step 1: Identify the Given Planes and Intersection

We are given two planes: \(l x + m y = 0\) and \(z = 0\). The plane \(z = 0\) is the xy-plane. The line of intersection of these two planes is the line \(l x + m y = 0\) in the xy-plane.

Step 2: Understand Plane Rotation

The plane \(l x + m y = 0\) is rotated about its line of intersection with the xy-plane through an angle \(\alpha\) with the plane remaining vertical to the xy-plane.

Step 3: Determine the Resultant Plane Equation

After rotating the plane through an angle \(\alpha\), we form a new plane. The new plane's equation will incorporate not only the original terms of the plane but also a term involving \(z\), adjusted by the angle \(\alpha\).

Step 4: Apply Trigonometry to the New Equation

Using trigonometry, the rotated plane's equation introduces a \(z\) term due to the rotation angle. The adjustment for \(z\) is \(\pm z \sqrt{l^2 + m^2} \sin \alpha\), derived from the sine of the angle of rotation and the original coefficients' magnitude. This makes option (A) the correct answer.

Key Concepts

Plane Rotation

Rotation in 3D geometry can be thought of as turning an object around a fixed line or axis. In the context of planes, imagine a flat surface being spun around a straight line that it touches. Here, the plane defined by the equation \(l x + m y = 0\) is rotated around its line of intersection with another plane, specifically \(z = 0\), which is the xy-plane.

During rotation, the line of intersection acts as a hinge or pivot. This transformation occurs while maintaining the plane's orientation relative to the intersecting line, almost as if the plane were being turned like a page around this axis.

• The concept of a plane rotation is crucial for visualizing how three-dimensional shapes can be manipulated while keeping certain boundaries fixed.
• Rotation maintains the original proportions and angles of the plane, except for its orientation in space, which changes.

This understanding provides a foundation for predicting how a plane's equation will adapt upon rotation.

Angle of Rotation

The angle of rotation, denoted by \(\alpha\), is the measure of the turn applied to the plane around its line of intersection with another plane. Specifically, this angle dictates how far the plane rotates from its initial position.

Visualize the angle \(\alpha\) as the degrees through which you twist the structure around the line of intersection. This angle is essential because it directly affects how the new plane's equation will be constructed after rotation.

• The sine of the angle of rotation, \(\sin \alpha\), plays a pivotal role in determining the adjustments made to the original equation to yield the new plane's characteristics.
• Understanding how trigonometric functions like sine interact with rotations allows for precise calculations necessary in defining new positions.

This focus on the angle \(\alpha\) helps to frame the transformation within the bounds of the problem's geometry.

Equation of a Plane

In geometry, the equation \(l x + m y = 0\) describes a plane's position in three-dimensional space. An equation of a plane represents all the points that lie flat on this geometric surface.

When a plane is rotated, its new position needs to be mathematically described. The adjustments involve not only the \(l\) and \(m\) terms but also an additional term related to \(z\), reflecting the spatial change due to rotation.

• In our rotated plane, this new equation is expressed as \(l x + m y \pm z \sqrt{l^2 + m^2} \sin \alpha = 0\).
• The \(\pm z \sqrt{l^2 + m^2} \sin \alpha\) term indicates the influence of the rotation angle and gives insight into the plane's new orientation in 3D space.

The plane’s equation post-rotation reveals how it is repositioned while maintaining its fundamental linear characteristics.

Line of Intersection

The line of intersection where two planes meet is a vital concept in 3D geometry. It acts as a fulcrum around which rotations and transformations are centered. In our example, where \(l x + m y = 0\) intersects \(z = 0\), this line is crucial to the rotated plane's behavior.

This intersection line lies on the xy-plane and can be visualized as the axis along which changes occur when \(\alpha\) is applied.

• The line serves as a "balance point," ensuring that the aspects of the plane's geometry remain constant throughout the transformation.
• It provides a reference frame for implementing the rotation angle, allowing us to redefine the plane's equation post rotation.

Understanding this line's role helps clarify how and why the plane's equation is altered after undergoing rotation.