Normals to a Quadratic through a Given Point

Jon Ingram
If we specify a point, like (4,4), how many normals to are there that go through that point?
1. At the start the specified point is (4, 3.5), and there appear to be two normals: one starting from (-1, 1), and the other from (2, 4). Verify that the two normals to through these two points do indeed go through (4, 3.5). Can you prove that there aren't any more? 2. Now move the point to (-1, 2). Prove that the normal through (1,1) goes through (-1, 2), and find the equations of the other two normals. 3. Now move the point to (2, 0.5). Find the equation of the single normal that goes through that point. Investigation Moving the point around, it appears that there are at most three normals to the curve that go through a given point. It also appears that there is a region which gives you three normals, and a region which gives you one normal. In particular, it appears that no point with a negative y-value generates three normals. Can you prove this? Investigate the algebra that was used to generate this applet, and prove that there can be at most three normals (and that there always has to be one). HARD: Can you find the equation of the boundary between the 'one normal' region and the 'three normal' region?