GATE 2023 ME – Question 32
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 $\pi = 3.14$.
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Correct answer: 18.75 to 18.93
Explanation
The transformation matrix is $\begin{bmatrix} 0 & 3 \\ 2 & 0 \end{bmatrix}$ with the determinant $0 - 6 = -6$. A linear map multiplies every area by $|\det| = 6$. The disc has area $\pi = 3.14$, so the transformed region (an ellipse) has area $6 \times 3.14 = 18.84$.