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February 26th, 2011, 09:14 PM   #1
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Tangent

Let 0 < ? < ? < 90.
If the quadratic equation x - 3ax + 1 - 4a = 0 has to have two roots tan ? and tan ?, determine the value of a. Also determine the value of

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February 27th, 2011, 08:44 PM   #2
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What are the answers supposed to be?
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February 27th, 2011, 10:32 PM   #3
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February 28th, 2011, 01:09 AM   #4
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Re: Tangent

The discriminant is 9a - 4(1 - 4a) = 9a + 16a - 4.
For this to be positive, a must be outside the interval (-810)/9.
So a < -2 or a > 2/9.

But the roots of the equation are both positive, so a > 0 and 3a > sqrt(9a + 16a - 4)
Solving, 9a > 9a + 16a - 4 so a < 1/4.

Finally, tan ? + tan ? = 3a, and tan ? tan ? = 1 - 4a. Therefore tan (?+?) = 3a/4a = 3/4.
Applying the double-angle formula for tan, and solving 3/4 = 2t/(1-t) for t, you find that t is either -3 or 1/3.
But t is positive, so t = 1/3.
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March 1st, 2011, 07:31 AM   #5
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Thanks aswoods!
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