Given 2D data with features X and Y and a circular decision boundary, which synthetic feature value should you add so a linear classifier can separate the classes?

A circular decision boundary implies that the distance from the origin (or a central point) determines class membership. A linear classifier can only create straight lines or hyperplanes. To enable a linear classifier to separate data with a circular boundary, you need to transform the data so that the circular pattern becomes linearly separable in a higher-dimensional space. The feature X2+Y2 represents the squared distance from the origin. If you add this as a new feature, a linear classifier can then draw a hyperplane based on this new feature's value, effectively creating a circular boundary in the original 2D space. X2 and Y2 alone would only allow for parabolic or linear boundaries along one axis. cos(X) introduces a periodic, non-linear transformation that doesn't directly address the circular separability problem.