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November 25th, 2009, 10:07 PM  #1 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Find equation of the common tangent to the parabolas
Find equation of the common tangent to the parabolas y^2 = 4ax and x^2 = 4by

November 26th, 2009, 01:22 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,114 Thanks: 2329 
The tangent to the parabola y² = 4ax at the point (at², 2at) has equation ty = x + at². The tangent to the parabola x² = 4by at the point (2bu, bu²) has equation ux = y + bu². If the above tangents have the same equation, tu = 1 and bu² = at, so u³ = a/b. Hence u = ³?(a/b) and t = ³?(b/a), where "³?" specifies the real cube root. 
December 1st, 2009, 09:19 PM  #3  
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Re: I couldnt understand the last 2 lines Quote:
Please could u explain me how did u get tu = 1 , bu^2 = at from the eg. of tangents....  
December 1st, 2009, 11:13 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,114 Thanks: 2329 
I multiplied the second equation by t, then compared it to the first equation. One term is already the same, and the equations I gave follow if the other two terms are the same.


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common, equation, find, parabolas, tangent 
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