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October 26th, 2014, 12:26 AM  #1 
Member Joined: Sep 2014 From: Cambridge Posts: 57 Thanks: 4  elliptic curve symmetry and derivative
Hi, I was given the formula: y^2 = x^3 +ax +b ; where a and b are constants. For the first question, it asks me to prove that the elliptic curve is symmetric about the xaxis. I drew a diagram and my answer is that all points on the curve are the same total distance away from both foci. Therefore, if (x,y) is flipped about the x axis, it becomes (x, y) and it is also on the curve, and the same distance away from both foci. For the second question, it asks me to find dy/dx for the elliptic curve formula provided. I come to dy/dx = (3x+a) / (2y) Are my answers correct? If not how can I approach this differently? 
October 27th, 2014, 06:59 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 492 
First question: for each x there are two values of which are equal in magnitude and opposite sign. Second question: the answer is incomplete. You should substitute the formula for y into the expresion, noting that there are two values, so the derivative has two values  not surprising since the curve has two branches. 
October 27th, 2014, 08:13 PM  #3 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,555 Thanks: 597 Math Focus: Wibbly wobbly timeywimey stuff.  

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