MT1100 Problem Sheet 5 Due: Thursday 12th November 2020 at 10:00. Please write your name clearly at the top of each page. Scan your work as a single-file PDF. Do not use too high a resolution to ensure the file size remains below 20 MB (the upper limit is 50MB). Then upload your PDF file onto Moodle, as per the instructions available from the General Information box on Moodle. You are encouraged to leave your handwritten comments to the marker on the first or last page of your homework submission. You have two weeks to complete this sheet, because of revision week. This problem sheet is all about standard forms of conics. 1. 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). 2. Show that 7x2 + 613 xy + 13y2 - 16 = 0 is the equation of an ellipse E. By rotating the axes anticlockwise through an angle of Ø = T/3 show that it may be written as (2+)+ (3) = 1. [Reminder: cos(T/3) = 1/2 and sin(T/3) = v3/2.] Hence sketch E (in the (x, y)-plane).