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# Set T consists of all points (x, y) such that x^2+y^2=1. If

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Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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15 Apr 2005, 05:45
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Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:
[Reveal] Spoiler: OA

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0012.jpg [ 9.03 KiB | Viewed 8839 times ]

Last edited by Bunuel on 21 Feb 2012, 00:35, edited 2 times in total.

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Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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21 Feb 2012, 00:36
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mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.
[Reveal] Spoiler:
Attachment:

graph.php.png [ 15.81 KiB | Viewed 8948 times ]

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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21 Feb 2012, 12:25
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anuu wrote:
Hi Bunnel,

Can you pls explain how did u calculate area of segment above the line (pi-2/4) for x^2+y^2<1

Anu

Sure. Look at the diagram. The area above the line equals to 1/4th of the area of the circle ($$\frac{\pi{r^2}}{4}$$) minus the area of the isosceles right triangle made by radii ($$\frac{1}{2}*r*r$$): $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi}{4}-\frac{1}{2}=\frac{\pi-2}{4}$$ (since given that $$r=1$$).

Hope it's clear.
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Re: PLS DRAW & ILLUSTRATE [#permalink]

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21 Feb 2012, 12:01
Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

Hi Bunnel,

Can you pls explain how did u calculate area of segment above the line (pi-2/4) for x^2+y^2<1

Anu

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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24 Mar 2012, 11:39
anuu wrote:
Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

Hi Bunuel,

Can you please take some time and clarify my doubt?

How did you arrive at P = 1/4, The portion of the circle that is above the line is (pi*r^2/4) - 1/* r^2 correct?

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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24 Mar 2012, 14:22
pkonduri wrote:
Hi Bunuel,

Can you please take some time and clarify my doubt?

How did you arrive at P = 1/4, The portion of the circle that is above the line is (pi*r^2/4) - 1/* r^2 correct?

Check this: set-t-consists-of-all-points-x-y-such-that-x-2-y-2-1-if-15626.html#p1047717

Hope it helps.
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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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20 Oct 2013, 13:16
Hello from the GMAT Club BumpBot!

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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26 Nov 2013, 05:48
Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

Great explanation! very elegant. thanks Bunuel.
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Re: PLS DRAW & ILLUSTRATE [#permalink]

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28 Dec 2013, 05:34
Bunuel wrote:
pkonduri wrote:
Hi Bunuel,

Can you please take some time and clarify my doubt?

How did you arrive at P = 1/4, The portion of the circle that is above the line is (pi*r^2/4) - 1/* r^2 correct?

Check this: set-t-consists-of-all-points-x-y-such-that-x-2-y-2-1-if-15626.html#p1047717

Hope it helps.

Hi Bunuel,

Still don't get how did you arrive at the conclusion that the area above the line within the circle is 1/4th of the circle for the original question

Would you kindly elaborate on this

Thanks
Cheers!
J

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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28 Dec 2013, 06:22
jlgdr wrote:
Bunuel wrote:
pkonduri wrote:
Hi Bunuel,

Can you please take some time and clarify my doubt?

How did you arrive at P = 1/4, The portion of the circle that is above the line is (pi*r^2/4) - 1/* r^2 correct?

Check this: set-t-consists-of-all-points-x-y-such-that-x-2-y-2-1-if-15626.html#p1047717

Hope it helps.

Hi Bunuel,

Still don't get how did you arrive at the conclusion that the area above the line within the circle is 1/4th of the circle for the original question

Would you kindly elaborate on this

Thanks
Cheers!
J

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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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28 Dec 2013, 07:10
OK sorry now i get it. For a second i forgot that pi = 3.14159

Thanks
Cheers
J

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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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25 May 2014, 03:58
Is that really a question which can come up in the GMAT? I believe the mathematics involved go well beyond! The MGMAT for example do not cover circles in xy planes at all.

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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08 Jun 2014, 21:42
Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

Hi Bunuel,

Do we have more variations of such questions. If yes, could you share it i want to practice.

R/
Ammu

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Re: PLS DRAW & ILLUSTRATE [#permalink]

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09 Jun 2014, 01:50
Expert's post
8
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BOOKMARKED
ammuseeru wrote:
Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

Hi Bunuel,

Do we have more variations of such questions. If yes, could you share it i want to practice.

R/
Ammu

I searched our questions banks and was able to find the following questions.

PROBABILITY AND GEOMETRY:

in-the-xy-plane-a-triangle-has-vertexes-0-0-4-0-and-88395.html
in-the-coordinate-plane-rectangular-region-r-has-vertices-a-104869.html
a-5-meter-long-wire-is-cut-into-two-pieces-if-the-longer-106448.html
a-triangle-with-three-equal-sides-is-inscribed-inside-a-160874.html
a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html
an-x-y-coordinate-pair-is-to-be-chosen-at-random-from-the-146005.html
a-cylinder-has-a-base-with-a-circumference-of-20pi-meters-132132.html
a-circular-racetrack-is-3-miles-in-length-and-has-signs-post-106203.html
in-the-coordinate-plane-rectangular-region-r-has-vertices-a-104869.html
in-the-xy-plane-the-vertex-of-a-square-are-88246.html
a-searchlight-on-top-of-the-watch-tower-makes-3-revolutions-76069.html
point-p-a-b-is-randomly-selected-in-the-region-enclosed-by-160615.html

Hope this helps.
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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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08 Dec 2014, 14:30
Hi Bunuel,

I understand the approach but can you please let me know how did you draw that blue line for f(x)=x+1?

Thanks

Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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08 Dec 2014, 14:34
sunita123 wrote:
Hi Bunuel,

I understand the approach but can you please let me know how did you draw that blue line for f(x)=x+1?

Thanks

Bunuel wrote:
mirhaque wrote:
Set T consists of all points (x, y) such that x^2+y^2=1. If point (a, b) is selected from set T at random, what is the probability that b>a+1?

(A) $$\frac{1}{4}$$
(B) $$\frac{1}{3}$$
(C) $$\frac{1}{2}$$
(D) $$\frac{3}{5}$$
(E) $$\frac{2}{3}$$

PLS DRAW & ILLUSTRATE:

Look at the diagram below.
Attachment:
graph.php.png

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

So, set T is the circle itself (red curve).

Question is: if point (a,b) is selected from set T at random, what is the probability that b>a+1? All points (a,b) which satisfy this condition (belong to T and have y-coordinate > x-coordinate + 1) lie above the line y=x+1 (blue line). You can see that portion of the circle which is above the line is 1/4 of the whole circumference, hence P=1/4.

If it were: set T consists of all points (x,y) such that $$x^2+y^2<1$$ (so set T consists of all points inside the circle). If point (a,b) is selected from set T at random, what is the probability that b>a+1?

Then as the area of the segment of the circle which is above the line is $$\frac{\pi{r^2}}{4}-\frac{r^2}{2}=\frac{\pi-2}{4}$$ so $$P=\frac{area_{segment}}{area_{circle}}=\frac{\frac{\pi-2}{4}}{\pi{r^2}}=\frac{\pi-2}{4\pi}$$.

Hope it's clear.

The blue line is y = x + 1. The area above it is y > x + 1. So, all the points for which y-coordinate is greater than x-coordinate + 1.
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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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15 Dec 2015, 11:44
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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If [#permalink]

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27 Jan 2017, 04:05
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Re: Set T consists of all points (x, y) such that x^2+y^2=1. If   [#permalink] 27 Jan 2017, 04:05
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