Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Two chords PA and PB divide circle into three parts. The angle PAB is a root of f(x)=0.

Find f(x)

Clearly , PA and PB divides circle into three parts means it divides it into 3 parts of equal areas

How can i find f(x) then ?

thanks

share|improve this question
    
I dont think equality of areas follows from the information you gave in your first sentence. I wonder whether you actually intended in your first sentence to specify that areas are to be equal. –  Michael Hardy Aug 2 '11 at 15:20
    
And once you find the angle $\theta$, there are an infinite number of functions that take $\theta$ to $0$. Hell, I could set $f(x)=0$ as a constant function without even thinking about the geometry. –  anon Aug 2 '11 at 15:36

2 Answers 2

You may assume your circle to be the unit circle in the $(x,y)$-plane and $P=(1,0)$. If the three parts have to have equal area then $A=\bigl(\cos(2\phi),\sin(2\phi)\bigr)$ and $B=\bigl(\cos(2\phi),-\sin(2\phi)\bigr)$ for some $\phi\in\ ]0,{\pi\over2}[\ $. Calculating the area over the segment $PA$ gives the condition $$2\Bigl({\phi\over2}-{1\over2}\cos\phi\sin\phi\Bigr)={\pi\over3}\ ,$$ or $f(\phi):=\phi-\cos\phi\sin\phi-{\pi\over 3}=0$. This equation has to be solved numerically. One finds $\phi\doteq1.30266$.

share|improve this answer

Hint: If you look up circular segment in Wikipedia, you should be able to write the area of the two segments cut off as a function of the angle between the chord and the tangent. That angle is $\theta/2$ using the figure in the article. Then the area of the circle that is left is what you want. The question is whether you can write this area as a function of the angle between the chords (one variable) instead of the angles between the chords and their respective tangents (two variables).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.