Problem Sheet 5 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 15 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. Let a and b be positive constants with b < a. Let C be the ellipse given by the equation 32 2 + 1= 1 . Remember that the eccentricity e of the ellipse is defined to be e = 1 - 2 Show that the distance from any point (x, y) on C to the focus (ae, 0) is equal to e times the distance from (x, y) to the directrix x = (1/e)a. (Hint: work with squares of distances.) 2. Let P be a point on the ellipse with eccentricity e and semimajor axis a. Let F and F' be the foci of the ellipse. Show that |PF|+|PF'| = 2a. Hint: The diagram below might help, where the vertical lines through N and N' are the directrices of the ellipse. Another hint: What is | NN'|? P N N O F F' 3. Bring the equation 16x2 - 9y2 + 96x + 36y - 36 = 0 into standard form. Hence show that this is the equation of a hyperbola H. Sketch H (in the (x, y)-plane), and find its centre and the equations of its two asymptotes (again, in the (x, y)-plane). 4. Show that 7x2 + 613 xy + 13y2 - 16 = 0 is the equation of an ellipse. By rotating the axes through an angle of Ø = T/3 show that it may be written as (21)2+ (2)2 4 == 1. [Reminder: cos(T/3) = 1/2 and sin(T/3) = v/3/2.] 2