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In a rectangular coordinate system, point A has coordinates [#permalink]
 20 Aug 2010, 14:30
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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}
Last edited by Bunuel on 19 Apr 2012, 11:53, 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]
20 Aug 2010, 14:48
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 Posted from my mobile device 
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Last edited by mainhoon on 20 Aug 2010, 16:05, edited 2 times in total.
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Re: What fraction of the area of circle C lies within ... [#permalink]
20 Aug 2010, 15:15
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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 [ 22.42 KiB | Viewed 3053 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]
23 Oct 2010, 13:33
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]
19 Apr 2012, 11:44
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]
19 Apr 2012, 12:00
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Re: What fraction of the area of circle C lies within ... [#permalink]
19 Apr 2012, 22:08
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 Posted from my mobile device Great work!!...Appreciate the stratgy.
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Re: In a rectangular coordinate system, point A has coordinates [#permalink]
08 Aug 2012, 08:57
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]
08 Aug 2012, 17:41
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]
23 Jan 2013, 10:15
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]
23 Jan 2013, 14:19
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]
04 Feb 2013, 09:37
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]
05 Feb 2013, 03:40
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]
05 Feb 2013, 05:50
Yea, thank you cleared things up
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Re: In a rectangular coordinate system, point A has coordinates [#permalink]
05 Feb 2013, 05:55
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Re: In a rectangular coordinate system, point A has coordinates
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05 Feb 2013, 05:55
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