Problem Sheet 6 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 22 November 2018 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1
1. Find the points that satisfy each of the following equations, and sketch the curve. [Look at the 'warning examples' I gave in lectures, where trying to put the equation into standard form fails.] (i) x2 + 4y2 - 8x - 8y + 20 = 0, (ii) x2 - y2 + x - y = 0. 2. For what values of k is the conic 3x2 + 5y2 - 18x + 4y = k (a) an ellipse, (b) a single point, (c) a curve with no points? 3. Consider the parabola y2 = 4ax. Take the origin of our polar coordinates to be at the focus (see the diagram below). Show that any point P on the parabola has polar coordinates (r, 0), where r = 2a/(1 - cos 0). [Hint: the diagram will help. Here MN is the directrix, and PMNQ is a rectangle.] -C M P r | N F ? - o Q 4. Show that the polar coordinates of the left-hand branch of the hyperbola in the notes satisfy r= a(e2-1)/(1+e cos 0). Hint: any point P on the left-hand branch satisfies | F2P| - | F1P] = 2a. 2