The matrix [50] 02 a) Draw the transformated shape. 5+ 4 3 2 1 -5 -4 -3 -2 -1 is applied to every point in the unit circle. Clear All Draw: -1 -2 -3 -4 -5 1 2 b) The area of the unit circle is T. 3 The area of the transformed shape is 4 5
The matrix [50] 02 a) Draw the transformated shape. 5+ 4 3 2 1 -5 -4 -3 -2 -1 is applied to every point in the unit circle. Clear All Draw: -1 -2 -3 -4 -5 1 2 b) The area of the unit circle is T. 3 The area of the transformed shape is 4 5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Matrix Transformation of the Unit Circle**
**Matrix Transformation:**
The matrix
\[
\begin{bmatrix}
5 & 0 \\
0 & 2
\end{bmatrix}
\]
is applied to every point in the unit circle.
**Task a):** Draw the transformed shape.
- A grid is provided with x and y coordinates ranging from -5 to 5.
- The transformation will affect the unit circle, altering its shape according to the given matrix.
**Explanation of Transformation:**
- The matrix \(\begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}\) scales the x-coordinate by 5 and the y-coordinate by 2.
- The unit circle (radius = 1) centered at the origin \((0,0)\) will transform into an ellipse.
**Task b):** Calculate the area of the transformed shape.
- **Original Unit Circle Area:** \(\pi\)
- **Transformed Shape (Ellipse) Area:**
- Use the formula for the area of an ellipse: \(\text{Area} = \pi \times a \times b\)
- Here, \(a = 5\) (x-axis scaling), \(b = 2\) (y-axis scaling).
- Transformed area = \(\pi \times 5 \times 2 = 10\pi\)
Fill in the answer box: The area of the transformed shape is \(10\pi\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9978ffa8-8e6a-4550-8363-e044b6a6e895%2F8815259f-b260-49dd-9f56-d40dc3592cb3%2Fjgmjtua_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Transformation of the Unit Circle**
**Matrix Transformation:**
The matrix
\[
\begin{bmatrix}
5 & 0 \\
0 & 2
\end{bmatrix}
\]
is applied to every point in the unit circle.
**Task a):** Draw the transformed shape.
- A grid is provided with x and y coordinates ranging from -5 to 5.
- The transformation will affect the unit circle, altering its shape according to the given matrix.
**Explanation of Transformation:**
- The matrix \(\begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}\) scales the x-coordinate by 5 and the y-coordinate by 2.
- The unit circle (radius = 1) centered at the origin \((0,0)\) will transform into an ellipse.
**Task b):** Calculate the area of the transformed shape.
- **Original Unit Circle Area:** \(\pi\)
- **Transformed Shape (Ellipse) Area:**
- Use the formula for the area of an ellipse: \(\text{Area} = \pi \times a \times b\)
- Here, \(a = 5\) (x-axis scaling), \(b = 2\) (y-axis scaling).
- Transformed area = \(\pi \times 5 \times 2 = 10\pi\)
Fill in the answer box: The area of the transformed shape is \(10\pi\).
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