Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(A) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \sin \alpha=0\).

Step 1: Understand the initial plane and rotation

We start with the plane given by the equation \(l x + m y = 0\) lying on the \(z = 0\) plane. This means the plane is on the XY-plane, and does not have a \(z\) component initially. We are tasked with finding the equation for this plane after it is rotated through an angle \(\alpha\) around its line of intersection with another plane \(z = 0\). The line of intersection is along the line \(lx + my = 0.\)

Step 2: Analyze the rotation effect

When the plane is rotated about its line of intersection, the change occurs primarily in the \(z\) direction. The given rotation can introduce a new \(z\) component. The resolution of the plane's rotation into new spatial coordinates typically uses trigonometric functions with respect to the rotation angle \(\alpha\). Since the plane originally has no \(z\) component, the introduction of a \(z\) term is necessary. The most common terms introduced are associated with \(\sin \alpha\), \(\tan \alpha\), or \(\cos \alpha\).

Step 3: Use rotational transformation

During the rotation, the distance normal to the original plane becomes influenced by \(\alpha\) and the length of the normal vector associated with \(l x + m y = 0\). This normal vector has a length \(\sqrt{l^2 + m^2}\), and the change due to rotation \(\pm z\sin\alpha\sqrt{l^2 + m^2}\) is added to the initial plane's equation. This is because the plane, after rotation, can oriented both ‘upwards’ or ‘downwards’ with resepct to its original position, indicated by the plus-minus sign in \(z\).

Step 4: Combine the information

Therefore, combining these adjustments and factors, the new form of the plane equation after considering the rotation becomes \(l x + m y \pm z \sqrt{l^2 + m^2} \sin\alpha = 0.\) This corresponds to option (A) from the problem's multiple-choice answers.

Key Concepts

Rotation of Planes

When we talk about the rotation of planes, we're considering how a flat surface in a 3D space can pivot around a specific line. This line, known as the line of intersection, acts as a hinge or axis. In our case, the line along which the plane \(lx + my = 0\) rotates is the line of intersection with the plane \(z = 0\). This means the plane pivots in the space above and below, altering its position.
Understanding plane rotation helps in visualizing how 2D surfaces behave within a bigger 3D environment.

• This process is akin to opening and closing the cover of a book while keeping its spine fixed.
• Through rotation, a new dimension is introduced, affecting the plane's equation, especially if one of the dimensions was initially zero. 

Remember, a plane before rotation is static relative to its axis, but after rotation, the entire plane shifts around this fixed line, altering how we describe its position mathematically.

Trigonometric Functions in Geometry

Trigonometric functions like \(\sin, \cos,\) and \(\tan\) play a crucial role in describing how objects move or orient themselves in space. When rotating a plane, these functions help quantify the effects of rotation on each axis.
In our scenario, the sine function, \(\sin \alpha\), captures how the rotated plane introduces a new component, specifically in the \(z\)-direction.

• \(\sin \alpha\) corresponds to changes in height when looking from the plane's initial line of intersection.
• These functions allow breakdowns of complex rotations into understandable mathematical terms.

Using trigonometric principles, it's possible to predict and calculate how much the plane's edges rise or fall due to rotation. The "+" or "−" in the rotation's result accounts for bi-directional shifts you might see in this form of geometric transformation.

Line of Intersection

The line of intersection is where two planes "crash" or meet each other in 3D space. For our exercise, the given intersection is along the equation \(lx + my = 0\) simultaneously with \(z = 0\). This line acts as a pivot point for any rotation occurring.
It's vital when defining where and how a plane rotates because:

• It remains unchanged throughout any rotational transformation.
• Any rotation only affects the surface elements of the plane, not the axis.

Understanding the line of intersection is like understanding the root or core from which a plane's rotation branches out. As the reference axis, this line is a constant in an equation that might otherwise change drastically through geometrical manipulations.

Normal Vector in Planes

A normal vector is a perpendicular marker to a plane's surface that represents its orientation in the 3D space. In the equation \(l x + m y = 0\), the normal vector is defined by the components \((l, m, 0)\) since initially, there is no \(z\)-component. However, after rotation, we must adjust this understanding.
This rotation introduces a component in the \(z\) direction, which we denote using \(\pm z \sqrt{l^2 + m^2}\sin \alpha\).

• This portrays the resultant position of the vector post-rotation.
• Think of the normal vector as an arrow pointing out of the plane, it identifies how the plane slices through the 3D space.

A vector reflects the magnitude and direction in which the effect of rotation influences the overall orientation of the plane. Calculating its length using \(\sqrt{l^2 + m^2}\) gives insight into how strong or pronounced the movement along a specific path might be, emphasizing changes due to geometrical rotations.