# 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 in±nity 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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