In a rectangular coordinate system, point A has coordinates

Question

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.

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}

Bunuel · Accepted Answer

Look a the diagram: 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.

mainhoon · Answer

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 = (pi/2-pi/4+1/2)/(pi/2) = (pi/4+1/2)/(pi/2) = (pi+2)/2pi

Posted from my mobile device

vineet474 · Answer

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) =  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.

GMATD11 · Answer

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

Bunuel · Answer

Solution: in-a-rectangular-coordinate-system-point-a-has-coordinates-99510.html#p767285 OA: D.

rajathpanta · Answer

Hi, Cant we do this by calculating the area of the sector and then the area of the circle??? Pls help me understand!

SOURH7WK · Answer

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

CleanSlate · Answer

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 ..

PrashantPonde · Answer

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.

pharm · Answer

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 ?

Bunuel · Answer

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.

pharm · Answer

Yea, thank you cleared things up

Bunuel · Answer

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).

megha_2709 · Answer

why we are considering square for this question, its nowhere specified in the question. I do not get this at all, please help.

Regards
Megha

ytsejam · Answer

A quick look at the answer choices tells you that the ratio does not depend on the value of "d". Hence you can choose "smart numbers" for d - For example 2. Which makes the radius of the circle sqrt(2). The all we need to do is add the area of the semi circle with the area of the triangle to get the area in the first quadrant (refer the diagram by Bunuel).

Solve and the answer is D.

Solve and the answer is D.

ytsejam · Answer

If you drop perpendiculars to x & y axes from (d,d) you will get (d,0) and (0,d). These two points are "d" units away from the origin as well as from (d,d). Knowing that all sides are of the same length and that the 3 angles are 90 degrees, (we drew perpendiculars to the axes and the axes themselves are at a () degree angle) we can conclude that the figure is a square.

Hope it helps.

Nums99 · Answer

Hey we cannot do this because we can use sectors only when the angle formed by the arc is at the centre of the circle.

ashKing12 · Answer

the question is asking = (total area of circle in the 1st quad) / total area of the circle

what mistake you are doing is = (total bulge area not in the 1st quad) / total area of the circle.

you forgot to consider the area of the square which is also the part of the circle's area present in the 1st quad

In a rectangular coordinate system, point A has coordinates

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In a rectangular coordinate system, point A has coordinates

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