PCA of Clusters - Intuition behind role of Variance 
The image above shows data generated from two Gaussians, forming two conspicuous clusters. The data is now projected onto two axes, and the projected data on one of the axes has higher variance than the other (i.e. 17.37 is more than twice of 8.35). The book I took this image from, further adds that, If cluster structure
is present in the data, using the projection with the highest variance is likely to
preserve this structure.
However, I don't seem to understand the intuition behind this statement - even though it's evident that it is true, from the attached picture. Could someone please explain? Thank you!
P.S. I'm only a beginner in Machine Learning, and this is my first interaction with the CrossValidated SE community. Pardon me if I've not presented the question properly, and let me know if this forum is appropriate for more such questions!
 A: The statement assumes that if the data contains discrete clusters, the axis or principal component of greatest variance is likely to separate those clusters. In the plot you've shown, the two principal components exhibit variances of 17.37 and 8.35. When the original data is projected onto the first component, which accounts for 17.37/(17.37+8.35) = 68% of the variance of the data, the 2 clusters are observed in the projection.
Note that the statement that the projection of highest variance is likely to preserve cluster information is a rule of thumb and is not always true.  For example, if the spread of the clusters in the y-direction was increased by a factor of 3, the 2 clusters (left vs right) would still be obvious by eye, but the cluster structure would be visible in the second principal component rather than the first component. This is because the axis labelled B would become the first component due to its accounting for more variance if the vertical spread of the data increased.
