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

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

None of the given options match the final coordinates. Check the calculations.

Step 1: Reflection About Line

To reflect the point \((4, 1)\) about the line \(y = x - 1\), we consider the line's equation \(y = x - 1\) as \(y - x = -1\). Using the formula for reflection over a line defined by \(ax + by + c = 0\), the reflected point \((x', y')\) is found as:\[ x' = \frac{x - y + by - ax}{a^2 + b^2} + x, \]\[ y' = \frac{y - x + ax - by}{a^2 + b^2} + y. \]Here, \(a = -1\), \(b = 1\), \(c = -1\). Substitute these into the formulas to find the reflected point:\[ x' = \frac{4 - 1 + 1(1-4)}{1 + 1} + 4= \frac{2}{2} + 4 = 3, \]\[ y' = \frac{1 - 4 + 1(4-1)}{1 + 1} + 1= \frac{0}{2} + 1 = 1. \]Thus, the reflected point is \((3, 1)\).

Step 2: Translation

To translate the point \((3, 1)\) along the positive \(x\)-axis by 1 unit, we add 1 to the x-coordinate:New point: \((3+1, 1) = (4, 1)\).

Step 3: Rotation About the Origin

To rotate the point \((4, 1)\) by an angle of \(\frac{\pi}{4}\) around the origin in the anti-clockwise direction, we use the rotation matrix:\[x' = x\cos\theta - y\sin\theta \y' = x\sin\theta + y\cos\theta\]Substitute \(x = 4\), \(y = 1\), and \(\theta = \frac{\pi}{4}\):Using \(\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\), we have:\[ x' = 4 \times \frac{\sqrt{2}}{2} - 1 \times \frac{\sqrt{2}}{2} = 2\sqrt{2} - \frac{\sqrt{2}}{2} = \frac{4\sqrt{2} - \sqrt{2}}{2} = \frac{3\sqrt{2}}{2},\]\[ y' = 4 \times \frac{\sqrt{2}}{2} + 1 \times \frac{\sqrt{2}}{2} = 2\sqrt{2} + \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}.\]So the point after rotation is \(\left(\frac{3\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\).

Key Concepts

Reflection in Lines

Reflecting a point across a line involves creating a mirror image of the point with respect to the line. In this scenario, the point \(4, 1\) is reflected over the line given by \(y = x - 1\). To determine the reflected point, we use the reflection formula for the line \(ax + by + c = 0\), where the line equation can be rewritten as \(y - x = -1\), giving us \(a = -1\), \(b = 1\), and \(c = -1\). By substituting these values into the reflection formula:

• \(x' = \frac{x - y + by - ax}{a^2 + b^2} + x\)
• \(y' = \frac{y - x + ax - by}{a^2 + b^2} + y\)

You can calculate the new coordinates resulting in the point \(3, 1\) after reflection. The role of the line is to act as a boundary or axis, ensuring symmetry between the original and reflected point. It's as though the original point is flipped over, maintaining equal distance from the line as before reflection.

Translation of Points

Translation in coordinate geometry is the process of moving a point a certain distance in a given direction. For this exercise, the point \(3, 1\) is translated 1 unit along the positive \(x\)-axis. To translate a point involves adjusting its coordinates by adding or subtracting values depending on direction:

• Add to the \(x\)-coordinate if moving right, subtract if moving left.
• Add to the \(y\)-coordinate if moving up, subtract if moving down.

Here, since the translation is 1 unit along the positive \(x\)-axis, we simply add 1 to the \(x\)-coordinate of the point, resulting in the new point becoming \(4, 1\). Translation maintains the shape and orientation of geometric figures, effectively shifting them in the plane without rotation or reflection.

Rotation of Points

Rotation in coordinate geometry means turning a point around a fixed point, generally, the origin, by a specific angle. In our case, after translation, the point \(4, 1\) is rotated about the origin through an angle of \(\frac{\pi}{4}\) (45 degrees) in an anti-clockwise direction. The rotation transformations use the following matrix equations:

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

Using these equations and substituting \(x = 4\), \(y = 1\), and \(\theta = \frac{\pi}{4}\), along with the fact that \(\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\), we calculate the rotated point as \(\left(\frac{3\sqrt{2}}{2}, \rac{5\sqrt{2}}{2}\right)\). This has the effect of not just changing the position of the point on the plane, but also altering the orientation relative to the origin.