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What is the area of the region enclosed by lines y=x, x=−y,

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What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 06 Apr 2013, 10:17
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m22 q20

What is the area of the region enclosed by lines y=x, x=-y, and the upper crescent of the circle y^2+x^2=4 ?

A. \frac{\pi}{4}
B. \frac{\pi}{2}
C. \frac{3\pi}{4}
D. \pi
E. 4\pi
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Last edited by Bunuel on 07 Apr 2013, 03:37, edited 1 time in total.
Edited the question.
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 06 Apr 2013, 10:58
Rock750 wrote:
What is the area of the region enclosed by lines y=x, x=−y, and the upper crescent of the circle y^2+x^2=4 ?

A- Pi/4

B- Pi/2

C- 3Pi/4

D- Pi

E- 4Pi


y^2+x^2=4 is a circle with its center in the origin (0,0)
The lines y=x and y=-x intersect in (0,0) and form an angle of 90° between them.
The area of the circle is r^2PI=4PI, now we are looking for "area of the region enclosed by lines y=x, x=−y, and the upper crescent of the circle y^2+x^2=4", which can be found through this equation 4PI : 360=x : 90 (Tot area : Tot angle = x : angBetweenLines)
D x=PI
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 06 Apr 2013, 19:56
The lines y=x, x=−y intersecting at (0,0) and they are perpendicular to each other.
The circle y^2+x^2=4 is centered at (0,0) and has a radius of 2.
These two lines are dividing the circle y^2+x^2=4 into four equal segments.
Area of each segment = (PI*r^2) / 4 = (PI*2^2) / 4 = PI

Answer is D.
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 07 Apr 2013, 03:38
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m22 q20

What is the area of the region enclosed by lines y=x, x=-y, and the upper crescent of the circle y^2+x^2=4 ?

A. \frac{\pi}{4}
B. \frac{\pi}{2}
C. \frac{3\pi}{4}
D. \pi
E. 4\pi

The circle represented by the equation x^2+y^2 = 4 is centered at the origin and has the radius of r=\sqrt{4}=2.

Look at the diagram below:
Attachment:
m22-20.png
m22-20.png [ 16.59 KiB | Viewed 1775 times ]

We need to find the area of the upper crescent, so the area of the yellow sector. Since the central angle of this sector is 90 degrees then its area would be 1/4 of that of the circle (since circle is 360 degrees).

The area of the circle is {\pi}{r^2}=4\pi, 1/4 of this value is \pi.

Answer: D.
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 08 Apr 2013, 11:45
Hi Bunuel,
in the above problem, i understand that the circle is centered at the origin and has radius 2 but lines y=x, and x = -1 represents the half of the circle so the area enclosed should be half of the total area, I mean 4pi/2. i know, i am missing something, please clarify!
and How do I know that question asks the area of upper 1/4 of the area or How do i determine the 90 degree portion??
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 08 Apr 2013, 12:01
kamalahmmad1

Quote:
How do i determine the 90 degree portion


Two lines are perpendicular if their slopes are -ve reciprocal of each other. E.g. line 1 : y = mx and line 2: y = (-1/m)x then line 1 and line two are perpendicular

Here y = x and y = -x meet the above stated criteria for perpendicular lines

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 09 Apr 2013, 02:46
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kamalahmmad1 wrote:
Hi Bunuel,
in the above problem, i understand that the circle is centered at the origin and has radius 2 but lines y=x, and x = -1 represents the half of the circle so the area enclosed should be half of the total area, I mean 4pi/2. i know, i am missing something, please clarify!
and How do I know that question asks the area of upper 1/4 of the area or How do i determine the 90 degree portion??


The two lines are y=x and y=-x (x=-y), not x = -1. Both lines are shown on the diagram in my post.

Lines y=x and y=-x make 90 degrees.
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 11 Aug 2013, 09:18
To find the Area of a sector

Arc angle Area of sector
___________ = ______________
360 Area of circle


90 Area of sector
___ = _____________
360 4 Pi

Area of sector = 4 pi x 90/ 360 = pi
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 13 Aug 2013, 14:26
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Rock750 wrote:
m22 q20

What is the area of the region enclosed by lines y=x, x=-y, and the upper crescent of the circle y^2+x^2=4 ?

A. \frac{\pi}{4}
B. \frac{\pi}{2}
C. \frac{3\pi}{4}
D. \pi
E. 4\pi

...
My solution(with detail explanation):
This diagram is indispensable:
Attachments

line and circle.png
line and circle.png [ 42.47 KiB | Viewed 1095 times ]


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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink] New post 16 Aug 2014, 09:15
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Re: What is the area of the region enclosed by lines y=x, x=−y,   [#permalink] 16 Aug 2014, 09:15
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