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In a rectangular coordinate system, point A has coordinates

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In a rectangular coordinate system, point A has coordinates [#permalink] New post 20 Aug 2010, 13:30
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In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

A. \frac{\pi}{\pi+\sqrt{2}}

B. \frac{\pi}{\pi+1}

C. \sqrt{\frac{2}{\pi}}

D. \frac{\pi+2}{2\pi}

E. \frac{2\pi}{2\pi+1}
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 Apr 2012, 10:53, edited 1 time in total.
Edited the question, added the answer choices and OA
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 20 Aug 2010, 14:15
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Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
CS.jpg [ 22.42 KiB | Viewed 4731 times ]
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is C=\pi{r^2}=\pi;
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

Ratio of the are of this region to area of a circle is \frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}.

Answer: D.

Hope it's clear.
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 20 Aug 2010, 13:48
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Ok so I am typing this on my mobile with imagination... Imagine a square of side 1 and a circle circumscribed.. Qn is what is the area that is lying on the 2 bulging sides..
Area of square = 1
area of circle = pi/2

The area of 4 bulging sides = (pi/2-1)
Fraction of circle area inside 1st Q= [pi/2-1/2(pi/2-1)]/pi/2 = (pi/2-pi/4+1/2)/(pi/2) = (pi/4+1/2)/(pi/2) = (pi+2)/2pi

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Last edited by mainhoon on 20 Aug 2010, 15:05, edited 2 times in total.
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 23 Oct 2010, 12:33
Let us assume A to be (1,1).Then, distance from origin = sqrt 2 = Diagonal of square = Diameter of circle.

Area of square = 1 sq units and
Area of circle = Pi r^2 = Pi (sqrt 2 / 2)^2 = Pi / 2

Area of 4 bulging sides = (Pi/2) - 1 = (Pi - 2) / 2 and Area of 2 bulging side = (Pi - 2) / 4.

Therefore, required fraction = Area of 2 bulging side / Area of circle = [(Pi - 2) / 4] / (Pi/2) = Pi -2 / 2Pi.

But answer posted by Bunuel is : Pi + 2 / 2Pi and I take his explainations as absolute. Since I got a different answer....please explain where I am wrong.
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 19 Apr 2012, 10:44
Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 19 Apr 2012, 11:00
Expert's post
GMATD11 wrote:
Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong


Solution: in-a-rectangular-coordinate-system-point-a-has-coordinates-99510.html#p767285
OA: D.
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 19 Apr 2012, 21:08
mainhoon wrote:
Ok so I am typing this on my mobile with imagination... Imagine a square of side 1 and a circle circumscribed.. Qn is what is the area that is lying on the 2 bulging sides..
Area of square = 1
area of circle = pi/2

The area of 4 bulging sides = (pi/2-1)
Fraction of circle area inside 1st Q= [pi/2-1/2(pi/2-1)]/pi/2 = (pi/2-pi/4+1/2)/(pi/2) = (pi/4+1/2)/(pi/2) = (pi+2)/2pi

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Re: In a rectangular coordinate system, point A has coordinates [#permalink] New post 08 Aug 2012, 07:57
Hi, Cant we do this by calculating the area of the sector and then the area of the circle??? Pls help me understand!
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Re: In a rectangular coordinate system, point A has coordinates [#permalink] New post 08 Aug 2012, 16:41
es we can do it by taking any value for (d,d) and calculating the area deducting the sector.
Take (d,d) as 2,2. the dia meter becomes 2 root 2 and radius root2
so area of circle = 2pie
area of 2 sectors= 2*(90/360)*2 pie = pie
But have to add another half of square = 1/2 * 4 = 2
So area in 1st quadrant = 2pie-pie+2= pie+2
and area of circle = 2pie
So ratio= (2+pie)/2pie
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 23 Jan 2013, 09:15
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is C=\pi{r^2}=\pi;
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

Ratio of the are of this region to area of a circle is \frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}.

Answer: D.

Hope it's clear.



Hi,

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 23 Jan 2013, 13:19
1
This post was
BOOKMARKED
FTG wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is C=\pi{r^2}=\pi;
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

Ratio of the are of this region to area of a circle is \frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}.

Answer: D.

Hope it's clear.



Hi,

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..


The question asks for "fraction" or "proportion" of the circle. (e.g. say 1/2 or 3/5th of circle lies in the first quadrant)
Hence you need to take the ratio of Area (in 1st quadrant) to Total area to find the fraction.
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 04 Feb 2013, 08:37
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is C=\pi{r^2}=\pi;
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

Ratio of the are of this region to area of a circle is \frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}.

Answer: D.

Hope it's clear.


Quote:
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;


is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?
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Re: What fraction of the area of circle C lies within ... [#permalink] New post 05 Feb 2013, 02:40
Expert's post
pharm wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Answer-choises will come later.


Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is C=\pi{r^2}=\pi;
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;

Area of a circle without the red parts is C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2};

Ratio of the are of this region to area of a circle is \frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}.

Answer: D.

Hope it's clear.


Quote:
Area of a square is half of the product of diagonals, as diagonal equals to 2r=2, then S=\frac{2^2}{2}=2;


is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?


If I understand correctly you are asking about the area of a square: area_{square}=side^2=\frac{diagonal^2}{2}. This is a general formula for the area of any square.

Hope it helps.
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Re: In a rectangular coordinate system, point A has coordinates [#permalink] New post 05 Feb 2013, 04:50
Yea, thank you cleared things up
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Re: In a rectangular coordinate system, point A has coordinates [#permalink] New post 05 Feb 2013, 04:55
Expert's post
pharm wrote:
Yea, thank you cleared things up


Forgot to mention that the area of a rhombus is also equals to half of the product of diagonals: area_{rhombus}=\frac{d_1*d_2}{2}, where d_1 and d_2 are the lengths of the diagonals (or bh, where b is the length of the base and h is the altitude).
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Re: In a rectangular coordinate system, point A has coordinates [#permalink] New post 22 Aug 2014, 14:54
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