Classic Geometry

Geometry Level 3

There are two circles of radius one, which intersect at two points, with the centre of one lying on the circumference of the other. The area of the wedge shaped region common to both the circles can be expressed in the form $\Large\frac{a\pi-b\sqrt{b}}{6}$ , where a and b are co-prime, non-negative integers, and b is not divisible by the square of a natural number.

Find the value of $\large a^{b}+b^{a}$ .

The answer is 145.

2 solutions

Damiann Mangan
Mar 11, 2014

Let's say the middle point of both circles are $A$ and $B$ with $X$ and $Y$ as the point of both intersections.

We could easily see that both $ABX$ and $ABY$ is an equilateral triangle because line $AB, AX, AY, BX,$ and $BY$ have length $1$ .

From that point, we could calculate that the problem is area of four pizza (two $ABX$ and two $ABY$ ) minus the area of two triangles ( $ABX$ and $ABY$ ).

Finally, we have $\frac{4\pi-3\sqrt{3}}{6}$ , which makes $a^{b}+b^{a}=64+81=145$ .

Let, 1st circle's center is on the origin & 2nd circle's center is on (1,0) . Therefore, the equations of the circles are $x^{2}+y^2=1$ and $(x-1)^{2}+y^2=1$ . The common region is bounded by the points $(0,0), (\frac{1}{2},+-\frac{\sqrt{3}}{2}), (1,0)$ . As, you can see, area of this region can be divided by 4 equal parts. For 1 part, area is bounded by the curve $x=0$ to $x=\frac{1}{2}$ and the curve $y=\sqrt{1-(x-1)^2}$ and can be obtained by, $\int_0^\frac{1}{2}\!\sqrt{1-(x-1)^2} dx$ which is, $\frac{\pi}{6}-\frac{\sqrt{3}}{8}$ . Finally multiplying by 4 we get $\frac{4\pi-3\sqrt{3}}{6}$ . Thus finally, a=4 and b=3. Therefore, $4^3+3^4=145$

Al Imran - 7 years, 2 months ago

Venture Hi
Mar 11, 2014

The first circle's euation is x62+y^2=1 2ith its center at (0,0). The second circle is (x-1)^2+y^2=1 with its center at (1,0) where it is on the perimeter of circle 1. Find the 2 intersecting points of the two circles : x= 1/2 and y= sqrt(3)/2 and -sqrt(3/2).

Integrate y-sqrt(1-x^2) from sqrt(3)/2 and -sqrt(3)/2 . You should get something like sqrt3/4+ pi/3. Muliply by 2 to get the entire wedge. a =4 and b is 3.

Classic Geometry

The answer is 145.

2 solutions

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