I'm going to ignore the part where you want to plot a cartesian object with polar coordinates.
You can either use ndgrid or meshgrid to generate a coordinate grid. Then you need to google the equation for an ellipse, after which it should be obvious how to adjust it to return all points inside the ellipse, instead of only the points exactly on the array.
To satisfy your requirement of each pixel being entirely inside the ellipse you can use the corners of each pixel and mark only the pixels if all four corners are inside.
Feel free to post a follow-up question in a comment if your have trouble implementing a specific step.