Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The point \((2,3)\) undergoes the following three transformations successively (i) reflection about the line \(y=x\) (ii) translation through a distance 2 units along the positive direction of \(y\)-axis (iii) rotation through an angle of \(45^{\circ}\) about the origin in the anti- clockwise direction. The final coordinates of the point are (A) \(\left(\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (B) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (C) \(\left(\frac{1}{\sqrt{2}},-\frac{7}{\sqrt{2}}\right)\) (D) none of these

Step-by-Step Solution

Answer

The final coordinates of the point are (B) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\).

Step 1: Reflect about the line y=x

Reflecting a point about the line \(y = x\) swaps its coordinates. Given point \((2, 3)\), after reflection, it becomes \((3, 2)\).

Step 2: Translate along the y-axis

Translate point \((3, 2)\) two units along the positive direction of the \(y\)-axis. The new coordinates are \((3, 2+2) = (3, 4)\).

Step 3: Rotate 45° anticlockwise about the origin

To rotate the point \((3, 4)\) around the origin by \(45^{\circ}\), we use the rotation matrix: \[\begin{bmatrix}\cos 45^{\circ} & -\sin 45^{\circ} \\sin 45^{\circ} & \cos 45^{\circ}\end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & -1 \1 & 1\end{bmatrix}\]Multiply this matrix with the point:\[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & -1 \1 & 1\end{bmatrix}\begin{bmatrix}3 \4\end{bmatrix}= \frac{1}{\sqrt{2}}\begin{bmatrix}-1 \7\end{bmatrix} \]Thus, the new point after rotation is \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\).

Key Concepts

Transformations in Geometry

In geometry, transformations alter the position, orientation, and size of shapes or objects. These changes are carried out using various operations like translation, reflection, and rotation, each having its distinct rules. Transformations are crucial in geometric computations and visualizations, helping us understand symmetry and movement in space. The original shape before transformation is called the preimage, while the resulting shape after transformation is called the image. It is important to follow specific rules for each transformation to ensure the shape's properties, like angles and side lengths, remain consistent.

Reflection

Reflection is like looking into a mirror; this transformation flips a shape over a specified line. This line is often referred to as the axis or line of reflection. When reflecting a point like (2, 3) ad along the line (y = x), the coordinates are traded, resulting in (3, 2). Key points about reflections:

• Coordinates switch based on the line of reflection.
• Reflection over y = x swaps the x and y values.
• The shape’s size and angles remain the same, but its orientation changes.

Reflections help in creating symmetrical designs and understanding geometric patterns.

Translation

Translation is shifting a shape in a straight line without rotating or flipping it. Imagine sliding a piece of paper across a table without turning it; this is essentially what translation does. In our example, translating the point (3, 2) two units up means adding 2 to the y-coordinate, resulting in the point (3, 4). Understanding Translations:

• Changes position but not the shape’s orientation or size.
• A vector informs how far and in what direction to move.
• All points in the shape shift uniformly.

Translations are handy in placing objects precisely in designs or solving spatial problems.

Rotation

Rotation involves turning a shape about a fixed point, often the origin. Rotations imply movement around a circle, similar to turning a dial. For example, rotating point (3, 4) by 45° anti-clockwise, using a rotation matrix, changes its position. The formula or matrix involves sine and cosine calculations of the angle:\[\begin{bmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta\end{bmatrix} =\frac{1}{\sqrt{2}}\begin{bmatrix}1 & -1 \1 & 1\end{bmatrix}\]Applying this, the new coordinates are \(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\).Key Aspects of Rotations:

• The shape rotates around a point, called the center of rotation.
• Angles of rotation determine how much the shape turns.
• Rotations preserve the shape's size and structure but change its orientation.

Rotations offer insights into circular movements and are used in various fields, including computer graphics and engineering.