Problem 40 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(A B C D\) is a square whose vertices \(A, B, C\) and \(D\) are \((0,0),(2,0),(2,2)\) and \((0,2)\), respectively. This square is rotated in the \(X-Y\) plane with an angle of \(30^{\circ}\) in anti-clockwise direction about an axis passing through the vertex \(A\). The equation of the diagonal \(B D\) of this rotated square is (A) \(\sqrt{3} x+(1-\sqrt{3}) y=\sqrt{3}\) (B) \((1+\sqrt{3}) x-(1-\sqrt{2})=2\) (C) \((2-\sqrt{3}) x+y=2(\sqrt{3}-1)\) (D) none of these

Step-by-Step Solution

Answer

The answer is (D) none of these.

Step 1: Identify initial points and calculate the center of rotation

The vertices of the square are given in clockwise order as \( (0,0), (2,0), (2,2), (0,2) \). The diagonal \( BD \) will have endpoints \( B(2,0) \) and \( D(0,2) \). The center of rotation is the point around which the square is rotated, which in this case is at vertex \( A(0,0) \).

Step 2: Rotation matrix for 30° anti-clockwise

To rotate a point \((x,y)\) about the origin by an angle \( \theta \), we use the rotation matrix: \[ R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \]For \( \theta = 30^{\circ} \), we have: \[ R = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix} \]

Step 3: Rotate points B and D

Applying the rotation matrix to point \( B(2,0) \):- \( x' = 2 \cdot \frac{\sqrt{3}}{2} - 0 \cdot \frac{1}{2} = \sqrt{3} \)- \( y' = 2 \cdot \frac{1}{2} + 0 \cdot \frac{\sqrt{3}}{2} = 1 \)Thus, \( B' = (\sqrt{3}, 1) \).Applying the rotation matrix to point \( D(0,2) \):- \( x' = 0 \cdot \frac{\sqrt{3}}{2} - 2 \cdot \frac{1}{2} = -1 \)- \( y' = 0 \cdot \frac{1}{2} + 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \)Thus, \( D' = (-1, \sqrt{3}) \).

Step 4: Equation of line through points B' and D'

The slope \((m)\) of the line through \((x_1, y_1) = (\sqrt{3}, 1)\) and \((x_2, y_2) = (-1, \sqrt{3})\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\sqrt{3} - 1}{-1 - \sqrt{3}} \]Solve to find the slope:\[ m = \frac{\sqrt{3} - 1}{-1 - \sqrt{3}} = -\frac{1}{2} \]Using point-slope form \((y - y_1) = m(x - x_1)\):\[ y - 1 = -\frac{1}{2}(x - \sqrt{3}) \]Expand and simplify to find the line equation:\[ y - 1 = -\frac{1}{2}x + \frac{\sqrt{3}}{2} \]\[ y = -\frac{1}{2}x + \frac{\sqrt{3}}{2} + 1 \]\[ y = -\frac{1}{2}x + \frac{\sqrt{3}}{2} + \frac{2}{2} \]\[ y = -\frac{1}{2}x + \frac{\sqrt{3} + 2}{2} \]

Step 5: Compare with given options

None of the provided options matches the equation \( y = -\frac{1}{2}x + \frac{\sqrt{3} + 2}{2} \). Thus, the equation of the diagonal \( BD \) for the rotated square is not among the choices (A), (B), or (C). Therefore, the answer is (D) none of these.

Key Concepts

Rotation Matrix

A rotation matrix is a crucial tool in coordinate geometry when performing rotations around a point or axis. It is a matrix used to rotate points in the Cartesian coordinate plane. To perform a rotation by an angle \( \theta \) around the origin, a 2D rotation matrix is used:

• The matrix is given by \( R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \).
• This matrix rotates points in an anti-clockwise direction.

For example, rotating by \( 30^{\circ} \), we replace \( \theta \) with \( 30^{\circ} \). Thus, the matrix becomes:\[ R = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix} \].
Rotating a point \( (x,y) \) involves multiplying its coordinates by this matrix, yielding the new coordinates \( (x', y') \). This process helps visually and spatially manipulate geometric figures.

Equation of a Line

Creating the equation of a line is an essential skill in analytical geometry. This equation represents a straight line in the coordinate plane and connects two specific points.

• For two points \((x_1, y_1)\) and \((x_2, y_2)\) the slope (m) of the line determines its angle compared to the x-axis.
• The equation of a line in point-slope form is \((y - y_1) = m(x - x_1)\).

This form is particularly useful when you already know one point on the line and the slope. The equation can be further simplified or expanded as needed to match standard or required forms.
For instance, the equation simplifies to the slope-intercept form \( y = mx + b \), which is often preferred for straightforward situations or graphing.

Slope of a Line

The slope of a line is a measure of its steepness, indicating how much \( y \) increases per unit increase in \( x \). Calculating slope is a fundamental concept for understanding line behavior.

• Mathematically, the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• Positive slope represents an upward slant, while a negative slope indicates a downward slant.
• A zero slope means a horizontal line.

In our exercise, solving the slope between points \((\sqrt{3}, 1)\) and \((-1, \sqrt{3})\) involves substituting coordinates into the slope formula. Mastery of slope calculations assists in forming accurate line equations and enables better predictions of line orientations.

Square Rotation

Rotating a square involves using geometric transformations, specifically rotation. A square is a quadrilateral with equal side lengths and right angles that can be rotated while maintaining these properties.

• The square's vertices must be recalculated after a rotation, as their coordinates change.
• Applying the rotation matrix to each vertex finds the square's new position.

In the practice example, rotate the square's vertices about its vertex \( A \) by \( 30^{\circ} \).
This rotation may shift the orientation of features, like diagonals, that require computation to understand post-rotation relationships. Understanding square rotation reveals much about geometric predictability in coordinate transformations.

Coordinate Transformation

Coordinate transformation is changing point positions through a specified operation, keeping the structure while altering locations. This process is instrumental in visualizing geometric transformations.

• Transformations include rotation, translation, scaling, and reflection.
• In rotations, points move along circular arcs about a fixed point.

The example problem uses a rotation coordinate transformation, affecting every vertex of the square. Post-rotation coordinates \( B' \) and \( D' \) illustrate new vertex positions, which require calculation using the rotation matrix.
Such transformations help understand how objects move within a coordinate plane without altering their intrinsic properties like side lengths or angles.