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What is the area of the region enclosed by the graph of the equation

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Bunuel
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What is the area of the region enclosed by the graph of the equation [#permalink] New post   09 May 2019, 21:40
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What is the area of the region enclosed by the graph of the equation \(x^2 + y^2 = |x| + |y|\)?


(A) \(\pi+\sqrt{2}\)

(B) \(\pi+2\)

(C) \(\pi+2\sqrt{2}\)

(D) \(2\pi+\sqrt{2}\)

(E) \(2\pi+2\sqrt{2}\)
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nick1816
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Re: What is the area of the region enclosed by the graph of the equation [#permalink] New post   09 May 2019, 22:41
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When x and y are positive
equation of the curve is x^2+y^2-x-y=0
or (x-1/2)^2 +(y-1/2)^2=1/2
which is equation of the circle having center, C at (1/2,1/2) and radius=(1/2)^{1/2}
Area of this circle lie in the 1st quadrant =Area of semicircle+ Area of AOC+ Area of BOC
Area of this circle lie in the 1st quadrant = (pi*r^2)/2 + 0.5r^2+0.5r^2={(pi/2)+1}r^2={(pi/2)+1}/2
By symmetry Total area in each quadrant is same
Hence total area= 4* {(pi/2)+1}/2= pi+2

Bunuel wrote:
What is the area of the region enclosed by the graph of the equation \(x^2 + y^2 = |x| + |y|\)?


(A) \(\pi+\sqrt{2}\)

(B) \(\pi+2\)

(C) \(\pi+2\sqrt{2}\)

(D) \(2\pi+\sqrt{2}\)

(E) \(2\pi+2\sqrt{2}\)

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GMATmona07
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Re: What is the area of the region enclosed by the graph of the equation [#permalink] New post   25 Jan 2023, 09:30
Bunuel Please provide official solution to this problem.
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bethebest96
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Re: What is the area of the region enclosed by the graph of the equation [#permalink] New post   10 Mar 2023, 00:44
nick1816 ThatDudeKnows area of semi circle + triangle aoc + triangle boc . didnt understand after that .. pls explain
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Re: What is the area of the region enclosed by the graph of the equation [#permalink] New post   10 Mar 2023, 09:17
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bethebest96 wrote:
nick1816 ThatDudeKnows area of semi circle + triangle aoc + triangle boc . didnt understand after that .. pls explain


bethebest96

I wouldn't worry about this question, as I don't see it as the sort of question GMAC would ask...at least not with the answer choices provided...mayyyyyyybe if the answer choices were really spread out from each other and only one were in the ballpark, but even then, probably not.

That said, I'm sure your curiosity is getting the better of you and you're going to want to spend (waste? ;) ) valuable study time satisfying that curiosity, so here goes.

Point A is at (0,1) and point B is at (1,0). Triangle AOB therefore as an area 0.5*1*1 = 0.5.
The diameter of the semicircle is \(\sqrt{2}\), so the area of the semicircle is 0.25pi.
The graph provided by nick1816 only shows one quadrant...a mirror of that graph also applies to the other three quadrants, so the total area is 4*(0.5+0.25pi) = pi+2. Answer choice B.
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Re: What is the area of the region enclosed by the graph of the equation [#permalink]
10 Mar 2023, 09:17
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