Question

A linear transformation maps a point \((x, y)\) in the plane to the point \((\hat{x}, \hat{y})\) according to the rule \[ \hat{x} = 3y, \hat{y} = 2x. \] Then, the disc \(x^2 + y^2 \le 1\) gets transformed to a region with an area equal to ................ (Rounded off to two decimals)
Use \(\pi = 3.14\).

Hint:

For any linear transformation from \((x, y)\) to \((\hat{x}, \hat{y})\), the area scaling factor is simply the absolute value of the determinant of the transformation's matrix. This is a powerful shortcut that avoids having to find the explicit shape of the transformed region.

Answer 1

Step 1: Understanding the Concept:
A linear transformation can be represented by a matrix. When a region in a plane is transformed, its area is scaled by a factor equal to the absolute value of the determinant of the transformation matrix. This determinant is known as the Jacobian of the transformation.
Step 2: Key Formula or Approach:
1. Represent the linear transformation \(T(x,y) = (\hat{x}, \hat{y})\) as a matrix \(A\). \[ \begin{pmatrix} \hat{x}
\hat{y} \end{pmatrix} = \begin{pmatrix} a & b
c & d \end{pmatrix} \begin{pmatrix} x
y \end{pmatrix} \] 2. The change in area is given by the formula: \[ \text{Area}_{\text{transformed}} = |\det(A)| \times \text{Area}_{\text{original}} \] The determinant of the transformation matrix is the Jacobian of the transformation. \[ \det(A) = \frac{\partial(\hat{x}, \hat{y})}{\partial(x, y)} = \begin{vmatrix} \frac{\partial \hat{x}}{\partial x} & \frac{\partial \hat{x}}{\partial y}
\frac{\partial \hat{y}}{\partial x} & \frac{\partial \hat{y}}{\partial y} \end{vmatrix} \] Step 3: Detailed Calculation:
1. Find the transformation matrix: The given transformation is \(\hat{x} = 0x + 3y\) and \(\hat{y} = 2x + 0y\). The matrix \(A\) is: \[ A = \begin{pmatrix} 0 & 3
2 & 0 \end{pmatrix} \] 2. Calculate the determinant (Jacobian): \[ \det(A) = (0)(0) - (3)(2) = -6 \] 3. Find the area scaling factor:
The scaling factor for the area is \(|\det(A)| = |-6| = 6\). 4. Calculate the original area:
The original region is a disc \(x^2 + y^2 \le 1\). This is a unit circle centered at the origin. The area of this disc is: \[ \text{Area}_{\text{original}} = \pi r^2 = \pi (1)^2 = \pi \] 5. Calculate the transformed area: \[ \text{Area}_{\text{transformed}} = |\det(A)| \times \text{Area}_{\text{original}} = 6 \times \pi \] Using the given value \(\pi = 3.14\): \[ \text{Area}_{\text{transformed}} = 6 \times 3.14 = 18.84 \] Step 4: Final Answer:
The area of the transformed region is 18.84.
Step 5: Why This is Correct:
The method correctly uses the property of linear transformations that the area of a transformed region is the area of the original region multiplied by the absolute value of the determinant of the transformation matrix. The original area and the determinant were calculated correctly. The transformed region is an ellipse, but we don't need its equation, only its area.