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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]

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06 Apr 2013, 11:17
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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. $$4\pi$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 07 Apr 2013, 04:37, 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]

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06 Apr 2013, 11:58
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]

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06 Apr 2013, 20:56
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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

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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07 Apr 2013, 04:38
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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 [ 16.59 KiB | Viewed 5094 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 $$\pi$$.

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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08 Apr 2013, 12:45
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]

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08 Apr 2013, 13:01
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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

//kudos please, if the above explanation is good.
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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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09 Apr 2013, 03:46
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Expert's post
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]

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11 Aug 2013, 10:18
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]

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13 Aug 2013, 15:26
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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 [ 42.47 KiB | Viewed 4392 times ]

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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19 Dec 2014, 04:53
All the 3 parts i.e x=y; x=-y; and x^2+y^2=4 makes one forth of the circular region x^2+y^2=4 area of which is 4pi so the required area is pi. hope this one makes it clear.

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09 Oct 2015, 23:49
What is the area of the region enclosed by lines y=x, x=−y, and the upper crescent of the circle y^2+x2=4 ?
a.π/4
b. π/2
c. π/4
d. π
e.4π

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10 Oct 2015, 02:36
sajib2126 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.π/4
b. π/2
c. π/4
d. π
e.4π

Question corrected!

Circle is y^2+x^2=4 has radius 2. Area of circle = pi * 4
y=x is the line that passes through 1 and 3 quadrants which has 45 degree to x-axis.
y=-x is the line that passes through 2 and 4 quadrants which has 45 degree to x-axis.

The two lines, divides the circle into 4 parts.

Combining all the three, (question to find the area of the upper crescent) Area = 4 pi/4 = pi

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10 Oct 2015, 02:42
Why or How does Circle is y^2+x^2=4 has radius 2 ?

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10 Oct 2015, 02:58
sajib2126 wrote:
Why or How does Circle is y^2+x^2=4 has radius 2 ?

In the equation x^2+y^2=4

all the points (2,2)(-2,2)(0,2)(2,0) are on the circle. And these have the center as (0,0)

Another explanation:
Equation Of A Circle
(x - a)^2 + (y - b)^2 = r^2
where
a is the x co-ordinate of the centre of the circle
b is the y co-ordinate of the centre of the circle
r is the radius of the circle

Comparing the above equation with the equation (x-0)^2+(y-0)^2=4
r^2 = 4
r=2

center is (0,0)

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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10 Oct 2015, 05:56
sajib2126 wrote:
What is the area of the region enclosed by lines y=x, x=−y, and the upper crescent of the circle y^2+x2=4 ?
a.π/4
b. π/2
c. π/4
d. π
e.4π

1. Search for a question before posting. It might have already been discussed before.
2. Topic title should match the first few words of the question.

Topics merged.

Refer above for the solution.

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Re: What is the area of the region enclosed by lines y=x, x=−y, [#permalink]

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16 Oct 2016, 09:25
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Re: What is the area of the region enclosed by lines y=x, x=−y,   [#permalink] 16 Oct 2016, 09:25
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