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.

Let $A,B, C,D$ and $E$ be five points marked, in clockwise order, on the unit circle in the plane (with centre at origin). Let $α$ and $β$ be real numbers and set $f (P ) = αx + βy$ where $P$ is a point whose coordinates are ($x, y$). Assume that $f (A) = 10$, $f (B) = 5$, $f (C ) = 4$ and $f (D) = 10$. Which of the following are impossible?
(a) $f (E) = 2$
(b) $f (E) = 4$
(c) $f (E) = 5$

totally stuck on it. how can I able to solve this problem

share|improve this question
    
All caps on the internet is interpreted as shouting. Please don't shout. –  Rahul Jan 13 '13 at 4:43
    
sorry for that. I did not mean that. the caps lock was on and did not notice it. extremely sorry again. –  user57097 Jan 13 '13 at 4:54
add comment

3 Answers

Hint: For a constant $k$, the set of points where $f(P) = k$ is a straight line.

share|improve this answer
    
did not understand.will you explain please.thanks for your time. –  user57097 Jan 13 '13 at 4:56
add comment

I claim that none of the three values presented are possible. As Calvin Lin notes, the locus of points for which the equation takes on a particular value $c$ is a line. More specifically, if we have $$c = \alpha x + \beta y$$ then we have $$y = -\frac{\alpha}{\beta}x + \frac{c}{\beta}$$ Notice that different values of the function are obtained by translating a line of slope $-\frac{\alpha}{\beta}$ up and down.

It follows that $A$ and $D$ lie on such a line with intercept $\frac{10}{\beta}$. Moreover, the line $\overline{AD}$ acts as a divider: all lines with $c>10$ lies on one side of the line and all lines with $c<10$ lies on the other side.

Since $B$ and $C$ lies within arc $\widehat{AD}$, it follows that they lie on the same side of $\overline{AD}$ and the side containing $B$ and $C$ is the side for which $c<10$. Now $E$ is necessarily contained on the other side of $\overline{AD}$ and therefore we must have $f(E) > 10$.

share|improve this answer
add comment

Take the plane $z = \alpha x + \beta y$ and examine the $z$ values as they vary over the unit circle. The values change direction twice, and since there is a direction change between $B$ and $D$, the value of $f(E)$ must be greater than $10$.

share|improve this answer
add comment

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.