Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

If the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is rotated about centre in its own plane by \(90^{\circ}\) in clockwise direction then the point \((a \cos \theta, b \sin \theta\) ) becomes (A) \((a \cos \theta,-b \sin \theta)\) (B) \((b \sin \theta,-a \cos \theta)\) (C) \((b \sin \theta, a \cos \theta)\) (D) none of these

Step-by-Step Solution

Answer

The point becomes (B) \((b \sin \theta, -a \cos \theta)\).

Step 1: Understand the Problem

We have an equation of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), and we need to find what happens to a point \((a \cos \theta, b \sin \theta)\) when the ellipse is rotated by \(90^{\circ}\) clockwise around its center.

Step 2: Determine the Effect of Clockwise Rotation by 90 Degrees

A point \((x, y)\) rotated by \(90^{\circ}\) clockwise transforms to \((y, -x)\).

Step 3: Apply the Transformation to the Given Point

Apply the \(90^{\circ}\) clockwise rotation transformation on the point \((a \cos \theta, b \sin \theta)\). The new coordinates will be \((b \sin \theta, -a \cos \theta)\).

Step 4: Match the Transformed Coordinates to the Given Choices

From our calculation, the transformed coordinates \((b \sin \theta, -a \cos \theta)\) match option (B).

Key Concepts

Coordinate Transformation

Coordinate transformation is a fundamental concept in geometry that involves changing the perspective from which we view a particular point or shape. It allows us to convert coordinates from one system to another. This process is essential in numerous mathematical applications, from simple graphing to complex engineering simulations.

In the context of the ellipse given in the problem, we are transforming the coordinates of a point as the ellipse undergoes rotation. Essentially, this means changing the position of the point \( (a \cos \theta, b \sin \theta) \) according to a specific rule defined by the rotation. Coordinate transformation helps us understand how orientation changes can affect the properties or positions of geometric figures. Here, we use a transformation rule specific for rotation, which will be explained next.

Rotation of Axes

Rotation of axes is a specific type of coordinate transformation. When a shape, such as an ellipse, is rotated around a point (often the origin or center of the ellipse), the coordinates of every point on the shape are affected. This principle is crucial in determining how the spatial characteristics of the figure change.

In algebra, rotating a point \( (x, y) \) around the origin by 90 degrees clockwise results in new coordinates given by the transformation formula: \( (y, -x) \). For the given problem, applying this rotation to the point \( (a \cos \theta, b \sin \theta) \) will change its position to \( (b \sin \theta, -a \cos \theta) \). This transformation rule is derived from the basic properties of rotation, using trigonometric principles to shift the axes while maintaining the relative angles between them.

Trigonometric Functions

Trigonometric functions like sine and cosine play a pivotal role in geometry, particularly in transformations involving rotation. They help express the coordinates of points on curves like ellipses in terms of an angle, \( \theta \), which simplifies the process of analyzing rotations.

The cosine function relates to the x-coordinate of a point on a unit circle, while the sine function relates to the y-coordinate. Consider the point \( (a \cos \theta, b \sin \theta) \); here, \( \theta \) represents the angle a hypothetical line from the origin makes with the positive x-axis. By expressing points with trigonometric functions, it simplifies the rotation transformation, as the angle \( \theta \) can dictate the effect of rotation on coordinates.

• Cosine shifts the x-value, reflecting the change along the horizontal axis.
• Sine shifts the y-value, reflecting the change along the vertical axis.

This understanding is crucial for correctly applying transformations like those needed for rotating the ellipse in the provided exercise.