Multiple choice

In a triangle ABC, CD is the bisector of the angle C. If $\displaystyle \cos \frac{C}{2}$ has the value $\displaystyle \frac{1}{3}$ and length of $ CD=6$, then$\displaystyle \left ( \frac{1}{a}+\frac{1}{b} \right )$ has the value equal to

  1. $\displaystyle \frac{1}{9}$
  2. $\displaystyle \frac{1}{12}$
  3. $\displaystyle \frac{1}{6}$

  4. none
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A Correct answer
Explanation

Using the formula for the length of the angle bisector CD = (2ab cos(C/2)) / (a+b). Given CD=6 and cos(C/2)=1/3, we have 6 = (2ab * 1/3) / (a+b). This simplifies to 6 = (2ab) / (3(a+b)), so 18 = 2ab / (a+b), which means (a+b)/ab = 2/18 = 1/9. Thus 1/b + 1/a = 1/9.

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