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Analytical Geometry

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21 Dec 2008 20:49:50 IST

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540

Find the set.

None

A circle with center $O$ and radius $R$ is drawn on a paper, and $A$ is a given point inside the circle

with $OA = a$ . Fold the paper to make a point $A^\prime$ on the circumference coincident with point $A$ , then a

crease line is left on the paper. Find out the set of all points on such crease lines, as $A^\prime$ goes through

every point on the circumference.

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Comments (6)

Rohit

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22 Dec 2008 14:22:52 IST

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In what form the set will be??Any example,sir?

Tiger on prowl.......

Blazing goIITian

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22 Dec 2008 14:48:01 IST

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sir,if you are talking about the set of common points of all such cease lines,its obviously null set.

if i am not mistaken,please explain what the question demands...........

Anant Kumar

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22 Dec 2008 23:07:40 IST

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No, I am not talking about the common points of these crease lines. Rather I want the set of all the points in each of the crease lines. For example, if get a crease line L₁ and another one as L₂ then, the points on both these lines that are inside the circle are elements of the set I require.

Rohit

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22 Dec 2008 23:37:16 IST

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Sir,in the answer we'll get the coordinates of points or what??

Anant Kumar

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22 Dec 2008 23:41:43 IST

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Well, at least the set is a collection of points. So you may obtain the coordinates or simply the relation that their coordinates must satisfy.

Rajat Sen

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24 Dec 2008 13:16:49 IST

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Let the circle be : $x^{2}+y^{2} = R^{2}$

Suppose I want to fold the paper in such a way such that a point on the circle

$P(Rcos \theta ,Rsin \theta )$ lie on the point $A(x_{0},y_{0})$ .

So, the crease will be a line which is the perpendicular bisector af $AP$ , $X$ being the mid - point of $AP$ .

So, $X \equiv \left(\frac{x_{0}+Rcos \theta }{2},\frac{y_{0}+Rsin \theta }{2}\right)$

Now it is easy to find the equation of the crease which I am getting as :

$x(Rcos \theta - x_{0}) + y(Rsin \theta - y_{0}) = \frac{R^{2} - a^{2} }{2}$

Moreover we have the condition that : $x^{2}+y^{2} \leq R^{2}$ .

So, by varying $\theta$ we can obtain the points .

I am trying to obtain the conditions on $\theta$ explicitly.

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