M14-28

Question

What is the area of the smaller sector enclosed by the graphs of  x^2 + y^2 = 4  and  y = |x| ?

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

Bunuel · Answer

Official Solution:

What is the area of the smaller sector enclosed by the graphs of  x^2 + y^2 = 4  and  y = |x| ?

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

The equation  x^2 + y^2 = 4  represents a circle centered at the origin with a radius of  \sqrt{4} = 2 .
 
The graph of  y = |x|  is shown below in blue:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/m14-28.png 
 
The smaller sector enclosed by these graphs is the yellow sector located in the top part of the circle. Since the central angle of this sector is 90 degrees, its area is  \frac{1}{4}  of the circle's area (given that the circle has a total angle of 360 degrees).
 
The area of the circle is  \pi r^2 = 4\pi . Therefore, the area of the smaller sector is  \frac{1}{4}  of this value, which is equal to  \pi  square units.

Answer: C

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation.

La1yaMalhotra · Answer

I understood the explanation. But I don't understand how the central angle is 90 degree? Secondly what we would have done if we were not sure of the central angle? How w would have approached the question in that case?

Bunuel · Answer

y = x makes 45 degree angle with y-axis. Similarly y = -x makes 45 degree angle with y-axis. The angle between them is 90 degrees.

OreoShake · Answer

lines with equations such as - y=x or y= -x would run through all the quadrants, not just quadrants 1 and 2. since the equation here is y=|x|, the two lines will run through all 4 quadrants in total. this means there is a possibility of an enclosure in  every  quadrant. Bunuel, please explain why only the enclosure in quadrant 1 and 2 have been taken for calculation of area of smaller sector.

Bunuel · Answer

Notice that y = |x| means that y cannot be negative (y is equal to an absolute value of a number which is always non-negative), that's why y = |x| is as presented below:

Bunuel · Answer

nicolasnemtala · Answer

Can't figure out how you got the radius of √4. I thought you can't simply find the roots of y^2 + x^2 = 4

Bunuel · Answer

Circle on a plane

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
 (x-a)^2+(y-b)^2=r^2

https://gmatclub.com/forum/download/file.php?id=10337

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

If the circle is centered at the origin (0, 0) , then the equation simplifies to:
 x^2+y^2=r^2

24. Coordinate Geometry 
     Theory   
 Math: Coordinate Geometry 
 Beauty of Coordinate geometry 
 Dealing with Tangents on the GMAT 
     Questions   
 DS Questions 
 PS Questions

For more check:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope it helps.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

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