# Area traced by point P with given conditions on the locus of P.

A point P moves inside a triangle formed by A(0,0) , B(1,√3.) , C(2,0) satisfying PA<=1 ,PB<=1 and >PC<=1 .If the area bounded by the curve traced by P is equal to aπ/b , then find the minimum value of (a+b).

I have tried to consider P as (h,k) and then related the distances of AP , BP , CP According to the condition given in the question. Since only one option amongst AP<= 1,BP<= 1,CP<= 1 has to be satisfied , therefore I tried considering only one option at a time.

That is:

1) if AP<=1:

But then again , I am overwhelmed by the number of variables in the question since I need to find the locus of the point P and find the area of the curve traced by P. Can Anyone give me a hint on how to solve this problem?

• Have you noticed that triangle ABC is equilateral ? But you should say at once that the set of points M fullfilling PA<1 OR PB<1 OR PC<1. May 17, 2020 at 7:37

Traingle $$ABC$$ is equilateral, which means that all its angles have value $$\pi/3$$.
The curve traced by $$P$$ is made of three arcs of circle, one at each vertex delimitating the sixth of a disk with radius 1 (whose area is $$\pi$$.
Therefore the 3 areas totalize a value $$3 \times \pi/6=\pi/2=a\pi/b$$ with $$a=1$$ and $$b=2$$,
Consequence $$a+b=3$$, because all other solutions will be written $$k \pi/(2k)$$ providing a larger value $$k+2k=3k$$.