GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Oct 2019, 22:33 GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  In the xy-plane, region Q consists of all points (x,y) such that x^2 +

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 58390
In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

10 00:00

Difficulty:   25% (medium)

Question Stats: 77% (01:47) correct 23% (01:58) wrong based on 320 sessions

HideShow timer Statistics

Tough and Tricky questions: Coordinate Geometry.

In the $$xy$$-plane, region $$Q$$ consists of all points $$(x, y)$$ such that $$x^2 + y^2 \le 100$$. Is the point $$(a, b)$$ in region $$Q$$?

(1) $$a + b = 14$$

(2) $$a \gt b$$

Kudos for a correct solution.

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58390
Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

1
5
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.

In the $$xy$$-plane, region $$Q$$ consists of all points $$(x, y)$$ such that $$x^2 + y^2 \le 100$$. Is the point $$(a, b)$$ in region $$Q$$?

(1) $$a + b = 14$$

(2) $$a \gt b$$

Kudos for a correct solution.

Official Solution:

The first step is to figure out what region $$Q$$ represents. Let's consider the boundary of region $$Q$$ by ignoring the "less than" part: $$x^2 + y^2 = 100$$. This equation represents a circle in the $$xy$$-plane, centered on the origin, with a radius of 10 (the square root of 100). Thus, region $$Q$$ consists of all points on or inside this circle. We are asked whether point $$(a, b)$$ lies inside this region. We can rephrase the question by substituting $$a$$ for $$x$$ and $$b$$ for $$y$$: do the variables $$a$$ and $$b$$ always satisfy the inequality $$a^2 + b^2 \le 100$$?

Statement (1): INSUFFICIENT. Relatively quickly, we can find a point on the line $$a + b = 14$$ that does not satisfy the inequality. Choose $$a = 0$$. Then $$b = 14$$, and the sum of the squares is 196, which is greater than 100. Thus, in this case, $$(a, b)$$ would not fall within region $$Q$$.

However, can we find any point on or within the circle? If we make both $$a$$ and $$b$$ equal 7, then the sum of their squares is $$49 + 49 = 98$$, which is less than 100. We could also choose $$a = 8$$ and $$b = 6$$, which gives us the sum of squares $$64 + 36 = 100$$. Either case satisfies the inequality, and so $$(a, b)$$ in these cases would fall within region $$Q$$.

Statement (2): INSUFFICIENT. The condition that $$a$$ is greater than $$b$$ is not very restrictive. We can find points that meet this condition both inside and outside the circle. For instance, $$(1, 0)$$ is within the circle, but $$(101, 100)$$ is not.

Statements (1) and (2) together: INSUFFICIENT. The case from statement 1 in which both $$a$$ and $$b$$ equaled 7 is no longer valid, but the case of $$a = 8$$ and $$b = 6$$ still works. Thus, we have a point satisfying both statements that lies on the circle. In fact, some suitable points lie within the circle, such as $$(7.5, 6.5)$$. However, we can still find suitable points that lie outside the circle - for instance, $$(14, 0)$$.

_________________
General Discussion
Manager  Joined: 10 Sep 2014
Posts: 96
Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

2
1
I chose answer choice E.

So basically what we are working with is (a+b)(a-b) < 100

statement 1: not sufficient
If A is -7 and B is 21, then (a+b)(a-b) would be much larger than 100 - not in region Q
If A is 7 and B is 7, then (a+b)(a-b) would be 0 - yes in region Q

statement 2: not sufficient
If A is 40 and B is 30, then (a+b)(a-b) would be 700 and this is not less than or equal to 100 - not in region Q
If A is 10 and B is 5, then (a+b)(a-b) would be 75 which is less than 100 - yes in region Q

So now I am down to answer choices C and E to choose from.

Combining both statements:
If A is 8 and B is 6, then (a+b)(a-b) would be 28 - yes in region Q
If A is 21 and B is -7, then (a+b)(a-b) would be much larger than 100 - not in region Q

Manager  Joined: 22 Sep 2012
Posts: 123
Concentration: Strategy, Technology
WE: Information Technology (Computer Software)
Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

2
x^2+y^2 <= 100 represents a circular region.

Statement 1 : a + b = 14

If a = 14 , b= 0, the point lies outside the region
If a= 7 , b = 7, then the point lies within the circle.
Insufficient

Statement 2 : a > b

If a = 15 , b= 0, the point lies outside the region
If a = 2, b= 2, the point lies within the circle.
Insufficient

Combining both , we can say
If a = 14 , b= 0, the point lies outside the region
If a = 8, b= 6, the point is on the circle.

Still it is not conclusive whetherthe point is outside or inside the circle. Therefore E) is the answer
Intern  B
Joined: 21 Mar 2017
Posts: 49
Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

Bunuel wrote:
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.

In the $$xy$$-plane, region $$Q$$ consists of all points $$(x, y)$$ such that $$x^2 + y^2 \le 100$$. Is the point $$(a, b)$$ in region $$Q$$?

(1) $$a + b = 14$$

(2) $$a \gt b$$

Kudos for a correct solution.

Official Solution:

The first step is to figure out what region $$Q$$ represents. Let's consider the boundary of region $$Q$$ by ignoring the "less than" part: $$x^2 + y^2 = 100$$. This equation represents a circle in the $$xy$$-plane, centered on the origin, with a radius of 10 (the square root of 100). Thus, region $$Q$$ consists of all points on or inside this circle. We are asked whether point $$(a, b)$$ lies inside this region. We can rephrase the question by substituting $$a$$ for $$x$$ and $$b$$ for $$y$$: do the variables $$a$$ and $$b$$ always satisfy the inequality $$a^2 + b^2 \le 100$$?

Statement (1): INSUFFICIENT. Relatively quickly, we can find a point on the line $$a + b = 14$$ that does not satisfy the inequality. Choose $$a = 0$$. Then $$b = 14$$, and the sum of the squares is 196, which is greater than 100. Thus, in this case, $$(a, b)$$ would not fall within region $$Q$$.

However, can we find any point on or within the circle? If we make both $$a$$ and $$b$$ equal 7, then the sum of their squares is $$49 + 49 = 98$$, which is less than 100. We could also choose $$a = 8$$ and $$b = 6$$, which gives us the sum of squares $$64 + 36 = 100$$. Either case satisfies the inequality, and so $$(a, b)$$ in these cases would fall within region $$Q$$.

Statement (2): INSUFFICIENT. The condition that $$a$$ is greater than $$b$$ is not very restrictive. We can find points that meet this condition both inside and outside the circle. For instance, $$(1, 0)$$ is within the circle, but $$(101, 100)$$ is not.

Statements (1) and (2) together: INSUFFICIENT. The case from statement 1 in which both $$a$$ and $$b$$ equaled 7 is no longer valid, but the case of $$a = 8$$ and $$b = 6$$ still works. Thus, we have a point satisfying both statements that lies on the circle. In fact, some suitable points lie within the circle, such as $$(7.5, 6.5)$$. However, we can still find suitable points that lie outside the circle - for instance, $$(14, 0)$$.

Bunuel
I understood this method. Would you plz help me to do the same DS using graphic method? I have tried but couldn't.
walker You can also help, I believe
Non-Human User Joined: 09 Sep 2013
Posts: 13204
Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +  [#permalink]

Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: In the xy-plane, region Q consists of all points (x,y) such that x^2 +   [#permalink] 14 Oct 2018, 01:53
Display posts from previous: Sort by

In the xy-plane, region Q consists of all points (x,y) such that x^2 +

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  