Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(B) \((b \sin \theta,-a \cos \theta)\)

Step 1: Understanding Rotation Effects

When an object in a plane is rotated by \(90^{\circ}\) clockwise, the coordinates of any point \((x, y)\) on that object transform to \((y, -x)\). This is derived from the basic principles of rotation in Cartesian coordinates.

Step 2: Apply Rotation to Given Point

The initial point given is \((a \cos \theta, b \sin \theta)\). Applying the \(90^{\circ}\) clockwise rotation transformation, the new point becomes \((b \sin \theta, -a \cos \theta)\).

Step 3: Match Transformed Coordinates with Options

Examine the transformed coordinates \((b \sin \theta, -a \cos \theta)\) and check which option it matches:- Option (A): \((a \cos \theta,-b \sin \theta)\)- Option (B): \((b \sin \theta,-a \cos \theta)\)- Option (C): \((b \sin \theta, a \cos \theta)\) - Option (D): none of these.The coordinates match with Option (B).

Key Concepts

Understanding Coordinate Transformation

Coordinate transformation is a technique that helps you change the frame of reference for a set of points or shapes in a plane. Imagine you have a point or a shape like an ellipse, and you want to know how it looks from a different angle without physically moving it. This process involves changing the coordinates from one form to another.

• To perform a coordinate transformation, you apply specific mathematical rules and functions.
• These rules depend on the type of transformation, like rotation, translation, scaling, or reflection.
• In this exercise, we specifically use rotation, where the coordinates are adjusted to reflect a new orientation in the plane.

Understanding coordinate transformation is essential in various mathematical and scientific applications. Whether you're working with geometric shapes, graphs, or even engineering models, mastering these transformations enables you to analyze and interpret data more effectively.

Exploring Cartesian Coordinates

Cartesian coordinates are a common way to represent the position of points in a plane using an ordered pair \((x, y)\). This system, named after the mathematician René Descartes, allows for easy manipulation and visualization of geometric problems.

• The x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position.
• Points in the plane are given by coordinates, making it easy to understand where they are located.
• In our example, the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) describes an ellipse centered at the origin in Cartesian coordinates.

Using Cartesian coordinates allows for straightforward calculations in rotation problems, as we can easily apply transformation rules to the ordered pairs. This mathematical framework is foundational to geometry, calculus, and many branches of science and engineering.

Analyzing Rotation Effects

Rotation effects refer to the changes in the coordinates of a point or a shape as it is rotated around a central point. In a Cartesian coordinate system, understanding the implications of rotation is crucial, especially when dealing with geometric figures such as ellipses.

• A clockwise rotation by \(90^{\circ}\) follows specific rules to transform the coordinates of a point.
• The point \((x, y)\) becomes \((y, -x)\) after a \(90^{\circ}\) clockwise rotation.
• Applying this to the point \((a \cos \theta, b \sin \theta)\), it transforms into \((b \sin \theta, -a \cos \theta)\).

Understanding how rotation affects points in a plane allows you to manipulate and reorient shapes without physically moving them. This knowledge is fundamental in fields ranging from computer graphics to robotics and beyond, where precise manipulation of objects in space is required.