Exercise 6 When we want to represent a binary object in an image with an object that has a simpler shape, we can use a region that has the shape of

an ellipse with the same zeroth, first, and second moments. An ellipse can be defined by the equation (x/a)² + (y/B)² = 1, where a is the semi-major axis long the x-axis and ẞ is the semi-minor axis along the y-axis. (a) Prove that the minimum and maximum second moments of the region about an axis through the origin are π/4 a 3³ and 1/4 ẞ a³, respectively. The second moment of any region about an axis inclined at an angle y can be written in the form E = a sin²³y-b sin y cos y + c cos² y. (b) Compute the major and minor axes of an equivalent ellipse, which means an ellipse that has the same second moment about any axis through the origin.