Topic : Coordinate geometry-Circles
from: category_eng
1. '

Two circles of radius 2 are centered at (2,0) and at (0,2). What is the area of the intersection of the interiors of the two circles?

extbf{(A) } pi -2 qquad extbf{(B) } frac{pi}{2} qquad extbf{(C) } frac{pi sqrt{3}}{3} qquad extbf{(D) } 2(pi -...

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2. '

Which of the following describes the graph of the equation (x+y)^2=x^2+y^2?

mathrm{(A) } extrm{the,empty,set}qquad mathrm{(B) } extrm{one,point}qquad mathrm{(C) } extrm{two,lines} ...

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1.

You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is a quarter of the circle with radius 2, and the isosceles triangle is a right triangle. Therefore, the area of half the intersection is
frac{1}{4} 4pi - frac{1}{2}(2)(2) = pi - 2
That means the area of the whole intersection is oxed{mathrm{(D) } 2(pi-2)}
2.

Expanding the left side, we have

x^2+2xy+y^2=x^2+y^2Longrightarrow 2xy=0Longrightarrow xy=0Longrightarrow x = 0 extrm{ or } y = 0

Thus there are two lines described in this graph, the horizontal line y = 0 and the vertical line x=0. Thus, our answer is mathrm{(C) }.