t06 - xy = z 2 to the parabola yz = x 2 . 4. Using your...

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The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 6 1. Find the equations of the tangent lines to the following conics at the indicated points: ( i ) The circle x 2 - 2 xz + z 2 + y 2 - 4 yz = 0 at the point (3 : 2 : 1). ( ii ) The hyperbola x 2 - y 2 = z 2 at the points (5 : 3 : 4) and (1 : - 1 : 0). ( iii ) The parabola yz = x 2 + xz at the points (1 : 2 : 1) and (0 : 1 : 0). 2. Find a collineation which maps the hyperbola xy = z 2 to the unit circle x 2 + y 2 = z 2 . 3. Find a collineation which maps the hyperbola
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Unformatted text preview: xy = z 2 to the parabola yz = x 2 . 4. Using your results from the two previous questions, nd a collineation which maps the unit circle x 2 + y 2 = z 2 to the parabola yz = x 2 . For your collineation, which point on the unit circle maps to the point at innity on the parabola? (Note that the answer to this will depend on your particular choices of collineations in the previous questions.)...
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