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The questions in the image refer to Principal Component Analysis (PCA) and involve identifying the direction of projections based on ellipses representing data distributions in two-dimensional space.

### Question Analysis:

1. **Figure 1 Analysis:**
   - The ellipse is elongated along the \( x_1 \)-axis. This suggests that the direction of maximum variance (first principal component) is along the \( x_1 \)-axis. 

   **Answer:**
   - **Option a) \( x_1 \)-axis** is correct.

2. **Figure 2 Analysis:**
   - The ellipse is elongated at an angle between the \( x_1 \) and \( x_2 \) axes, indicating that the direction of maximum variance (first principal component) is along a diagonal direction.

   **Answer:**
   - **Option c) Direction making an angle of 45° with \( x_1 \)-axis in anticlockwise direction** is correct.

Would you like a more detailed explanation or have any other questions?

### Relative Questions:

1. What is Principal Component Analysis (PCA) and how is it used in data analysis?
2. How does the orientation of an ellipse indicate the direction of maximum variance in PCA?
3. What are the implications of projecting data onto the first principal component?
4. How does PCA reduce the dimensionality of data?
5. What is the significance of the eigenvectors in determining the directions of the principal components?

### Tip:
In PCA, the first principal component represents the direction of maximum variance, and each subsequent component represents the next highest variance orthogonal to the previous ones.

Which of the following could be the direction of projection using PCA if the data points lie on the locus of the curve shown in Figure 1 and Figure 2?

Explore how to determine the direction of projections using Principal Component Analysis (PCA) based on data represented as ellipses. Analyze Figures 1 and 2 to identify the correct projection angles using eigenvector decomposition and covariance matrix calculations.

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