Problem 27 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The point \((4,1)\) undergoes the following three successive transformations (A) Reflection about the line \(y=x-1\) (B) Translation through a distance 1 unit along the positive \(x\)-axis (C) Rotation through an angle \(\frac{\pi}{4}\) about the origin in the anti- clockwise direction. Then, the coordinates of the final point are (A) \((4,3)\) (B) \(\left(\frac{7}{2}, \frac{7}{2}\right)\) (C) \((0,3 \sqrt{2})\) (D) \((3,4)\)

Step-by-Step Solution

Answer

The final coordinates are (0, \(3\sqrt{2}\)).

Step 1: Reflection about the line y=x-1

The line equation is given as \( y = x - 1 \). The transformation rule for reflection over the line \( y = x - a \) is : for a point \((x_1,y_1)\), the reflected point becomes \((y_1 + a, x_1 - a)\). For the point \((4,1)\), the reflected point is at \( (1 + 1, 4 - 1) = (2, 3) \).

Step 2: Translation along the x-axis

The point resulting from Step 1 is \((2,3)\). A translation of 1 unit along the positive \(x\)-axis adds 1 to the \(x\) coordinate without changing the \(y\) coordinate. Thus, the point becomes \((2 + 1, 3) = (3, 3)\).

Step 3: Rotation about the origin

Now, we need to rotate the point \((3, 3)\) through an angle of \(\frac{\pi}{4}\) (45 degrees) about the origin. The transformation rule for a rotation about the origin is \[x' = x \cos(\theta) - y \sin(\theta) \y' = x \sin(\theta) + y \cos(\theta) \] where \(\theta\) is the rotation angle. Substituting: \[x' = 3 \cos\left(\frac{\pi}{4}\right) - 3 \sin\left(\frac{\pi}{4}\right) = 3\left(\frac{\sqrt{2}}{2}\right) - 3\left(\frac{\sqrt{2}}{2}\right) = 0 \y' = 3 \sin\left(\frac{\pi}{4}\right) + 3 \cos\left(\frac{\pi}{4}\right) = 3\left(\frac{\sqrt{2}}{2}\right) + 3\left(\frac{\sqrt{2}}{2}\right) = 3\sqrt{2} \] So, the final point is \((0, 3\sqrt{2})\).

Key Concepts

Transformation

In coordinate geometry, a transformation refers to the process of changing the position, orientation, or size of a shape in a plane. Transformations can be broken down into various types, such as reflections, translations, and rotations. Each transformation has its unique properties describing how it alters the points or shapes involved.

Transformations often involve rules or formulas that dictate how each point is moved. They can be applied sequentially to produce complex shifts in position. For example, starting with a point \( (4,1) \), we might apply a series of transformations, such as reflections, translations, or rotations, each affecting the coordinates in different ways. Understanding these principles allows us to predict the final position of a point after multiple transformations.

Transformations are vital in various applications, from computer graphics to solving geometric problems, helping us to maneuver shapes and patterns effortlessly.

Reflection

Reflection is a type of transformation that produces a mirror image of a point or shape over a specific line known as the line of reflection. To perform a reflection, each point of the figure is flipped across this line, maintaining equal distance from it but on the opposite side.

The standard procedure for reflecting a point \( (x_1, y_1) \) over the line \( y = x - a \) involves switching and adjusting the coordinates. The new coordinates become \( (y_1 + a, x_1 - a) \). Hence, for the given point \( (4, 1) \) across the line \( y = x - 1 \), we compute the reflected point as \( (2, 3) \).

• Key Point: Always ensure to identify the correct line of reflection to apply the right formula.
• Application: Use these reflections in solving symmetrical object problems or creating patterns.

Reflection keeps the shape and size of the original figure intact, only altering its position and orientation.

Translation

Translation is a simple transformation that slides a shape or point in the coordinate plane along a vector, without changing its orientation, size, or shape. In this exercise, we consider translation along the coordinate axes.

When translating a point, such as \( (2, 3) \), through a distance of 1 unit along the positive \( x \)-axis, we adjust the \( x \)-coordinate while keeping the \( y \)-coordinate constant. The result yields a new position \( (3, 3) \).

The essential quality of translation is that it preserves the orientation and shape of figures:

• Impact: Only the position changes; other characteristics remain preserved.
• Formula Insight: For a translation, simply add the distance value to the x or y coordinate as specified.

This fundamental concept supports the construction of new positions while retaining the inherent properties of the translated figure.

Rotation

Rotation involves turning a figure or point around a fixed point known as the center of rotation. In coordinate geometry, this is often the origin unless specified otherwise. The angle of rotation dictates how far and in which direction the figure rotates.

For a rotation about the origin by angle \( \theta \), the coordinate transformation uses:

• \( x' = x \cos(\theta) - y \sin(\theta) \)
• \( y' = x \sin(\theta) + y \cos(\theta) \)

In our problem, rotating point \( (3, 3) \) by \( \frac{\pi}{4} \) (or 45 degrees) involves these transformations, resulting in the final point \( (0, 3\sqrt{2}) \).

• Angle Specifics: Positive angles usually imply counterclockwise rotation.
• Preservation: Distances from the center remain unchanged; only the orientation shifts.

Through rotation, one can explore rotational symmetries and reposition figures effectively on the coordinate plane.