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

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 are (-1/√2, 7/√2).

Step 1: Reflect the point about the line y = x

To reflect the point \((2,3)\) about the line \(y = x\), we swap the coordinates. So, the reflection of \((2,3)\) is \((3,2)\).

Step 2: Translate the point along the positive y-axis

Next, we translate the reflected point \((3, 2)\) by 2 units along the positive y-axis. This involves adding 2 to the y-coordinate. Thus, the translated point is \((3, 4)\).

Step 3: Rotate the point 45 degrees anticlockwise about the origin

To rotate the point \((3, 4)\) 45 degrees anticlockwise about the origin, we use the rotation formulas: \(x' = x \cos(45^{\circ}) - y \sin(45^{\circ})\) and \(y' = x \sin(45^{\circ}) + y \cos(45^{\circ})\).\ Substituting the values, we use \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{1}{\sqrt{2}}\):\ \[x' = 3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} - \frac{4}{\sqrt{2}} = -\frac{1}{\sqrt{2}}\] \[y' = 3 \cdot \frac{1}{\sqrt{2}} + 4 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}} = \frac{7}{\sqrt{2}}\]\ Thus, the rotated point is \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\).

Key Concepts

Reflection Transformation

Reflection transformation in coordinate geometry is a way of mirroring a point across a specific line. Imagine it like looking in a mirror and seeing your reflection on the other side. When reflecting a point

• Across the x-axis: Only the y-coordinate changes its sign.
• Across the y-axis: Only the x-coordinate changes its sign.
• Across the line y = x: The x and y coordinates swap places.

For example, when we reflect the point \(2,3\) across the line \(y=x\), we switch the coordinates to get \(3,2\). This concept is crucial because it gives us symmetry in geometry, allowing us to predict positions after a reflection transformation.

Translation in Geometry

Translation in geometry refers to sliding a point or a shape to a new location without altering its orientation. Picture it as pushing an object along a table without rotating it. You move every point by the same distance in the same direction.
For translating:

• Along the x-axis: Add or subtract a certain value to the x-coordinate.
• Along the y-axis: Add or subtract a certain value to the y-coordinate.

In our exercise, after reflecting the point \(3,2\), we translated it 2 units up along the y-axis, resulting in \(3,4\). This translation step increases the y-coordinate by 2, leaving the x-coordinate unchanged. Understanding translation helps visualize how objects move across a plane without changing their appearance.

Rotation Transformation

Rotation transformation involves turning a point or a shape around a central point, usually the origin in coordinate geometry. Imagine spinning a piece of paper around a pin stuck through its center. The points move but stay equidistant from the center.
For rotation:

• Use angles measured in degrees or radians, with a common example being 45° or \(\pi/4\) radians.
• The rotation formulas are \(x' = x\cos\theta - y\sin\theta\) and \(y' = x\sin\theta + y\cos\theta\).

In our scenario, using an angle of 45°, the point \(3,4\) rotates around the origin to become \(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\). These transformations are essential for understanding how objects can be manipulated in space, crucial for both theoretical mathematics and practical applications such as graphics design or engineering.