Tangents to a Quadratic through a Given Point

Author:
Jon Ingram
Given a point, like (1, -3), can you find the equations of the tangents to that go through that point?
1. Show that the tangents to through (-1,1) and (3,9) go through the point (1, -3). 2. Find the equations of the tangents that go through (2, 3). 3. Choose a point on the negative y-axis, and find the equations of the tangents that go through that point. 4. Choose any other point not in the grey region, and calculate the equations of the tangents through that point. Investigation 5. What is different about the grey region? Can you prove it? 6. In questions 1 and 2, the tangent points had 'nice' (integer) coordinates. Can you find other 'through points' which give nice tangent points? Can you find a general rule? 7. What is the general formula for the equations of the tangents through a point ?