Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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

Show Tags

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 \(x=PI\)
_________________

It is beyond a doubt that all our knowledge that begins with experience.

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

Show Tags

06 Apr 2013, 20:56

1

This post received KUDOS

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

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 \(\pi\).

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

Show Tags

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??

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

Show Tags

08 Apr 2013, 13:01

1

This post received KUDOS

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

//kudos please, if the above explanation is good.
_________________

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.
_________________

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

Show Tags

16 Aug 2014, 10:15

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

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

Show Tags

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.

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

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)

Hence the radius is 2.

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

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

Show Tags

16 Oct 2016, 09:25

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Version 8.1 of the WordPress for Android app is now available, with some great enhancements to publishing: background media uploading. Adding images to a post or page? Now...

Post today is short and sweet for my MBA batchmates! We survived Foundations term, and tomorrow's the start of our Term 1! I'm sharing my pre-MBA notes...

“Keep your head down, and work hard. Don’t attract any attention. You should be grateful to be here.” Why do we keep quiet? Being an immigrant is a constant...