February 26th, 2011, 10:14 PM  #1 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Tangent
Let 0° < ? < ? < 90°. If the quadratic equation x²  3ax + 1  4a = 0 has to have two roots tan ? and tan ?, determine the value of a. Also determine the value of 
February 27th, 2011, 09:44 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,442 Thanks: 1462 
What are the answers supposed to be?

February 27th, 2011, 11:32 PM  #3 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  
February 28th, 2011, 02:09 AM  #4 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Tangent
The discriminant is 9a²  4(1  4a) = 9a² + 16a  4. For this to be positive, a must be outside the interval (8±10)/9. So a < 2 or a > 2/9. But the roots of the equation are both positive, so a > 0 and 3a > sqrt(9a² + 16a  4) Solving, 9a² > 9a² + 16a  4 so a < 1/4. Finally, tan ? + tan ? = 3a, and tan ? tan ? = 1  4a. Therefore tan (?+?) = 3a/4a = 3/4. Applying the doubleangle formula for tan, and solving 3/4 = 2t/(1t²) for t, you find that t is either 3 or 1/3. But t is positive, so t = 1/3. 
March 1st, 2011, 08:31 AM  #5 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
Thanks aswoods!


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