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

It is currently 26 Jun 2019, 05:13

Close

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
Your Progress

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.

Close

Request Expert Reply

Confirm Cancel

Perpendicular lines m and n intersect at point (a, b), where a > b > 0

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

Hide Tags

 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55802
Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post 23 May 2019, 02:21
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

27% (03:03) correct 73% (02:40) wrong based on 51 sessions

HideShow timer Statistics


Perpendicular lines m and n intersect at point (a, b), where a > b > 0. The slope of line m is between 0 and 1. Which of the following statements MUST be true ?

I. The x-intercept of line n is positive.
II. The product of the x- and y-intercepts of line m is not positive,
III. The sum of the x-intercepts of lines m and n is positive.

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

_________________
Director
Director
avatar
P
Joined: 19 Oct 2018
Posts: 564
Location: India
Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post 23 May 2019, 06:19
Let equation of line m is y=kx+c
As line m passes through (a,b), hence b=ka+c or c=b-ka
Equation of line m, y=kx+(b-ka)

Similarly Equation of line n, y=(-1/k)x + (b+a/k)

1. Y- intercept of line n= (b+a/k)
as a,b >0 and 0<k<1
hence b+a/k will always positive

2. Y-intercept of line m= (b-ka)
X- intercept of line m= -(b-ka)/k
The product of the x- and y-intercepts of line m= -[(b-ka)^2]/k
Hence The product of the x- and y-intercepts of line m is always less than or equal to zero or can never be positive

3. X- intercept of line m= -(b-ka)/k
X- intercept of line n= (b+a/k)*k
Sum of X- intercept of line m and n= -(b-ka)/k + (b+a/k)*k= 2a+b(k-1/k)
a,b>0 but (k-1/k)<0, when 0<k<1
Hence Sum of X- intercept of line m and n can be positive or negative, depending upon the values of a,b and k

IMO D
GMAT Tutor
avatar
G
Joined: 24 Jun 2008
Posts: 1662
Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post 23 May 2019, 08:32
I'd just draw pictures here to see what can happen. Our two lines meet at a point in the first quadrant (where x and y coordinates are positive), at a point (a, b), where a > b, so where the x-coordinate exceeds the y-coordinate. That means our intersection point lies below the line y=x, but above the x-axis. Line m has a slope between 0 and 1, so as it moves right, it's rising, but slowly. The perpendicular line n, therefore, is falling and quickly as it moves to the right.

So, if line n has a point on it (a, b) above the x-axis in the first quadrant, and then is falling quickly as you move right from that point, it must hit the x-axis further to the right of that point, so its x-intercept must be positive, and I must be true.

If line m has a positive slope, independent of any other information provided, the product of its x and y intercepts will be negative (or zero if it passes through the origin). You can see that just considering two cases: if its x-intercept is negative, then it rises moving right from there, so has a positive y-intercept. If its x-intercept is positive, it falls moving left from there, so its y-intercept is negative. So II must be true.

III need not be true, because the x-intercept of line m could be negative one trillion if it has a very shallow slope.

I think there's a typo in the answer choices - I imagine answer D is meant to read "I and II only", and not "I and III only". (edit - that has since been fixed in the OP)
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Target Test Prep Representative
User avatar
D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 6679
Location: United States (CA)
Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post Updated on: 05 Jun 2019, 06:50
Bunuel wrote:
Perpendicular lines m and n intersect at point (a, b), where a > b > 0. The slope of line m is between 0 and 1. Which of the following statements MUST be true ?

I. The x-intercept of line n is positive.
II. The product of the x- and y-intercepts of line m is not positive,
III. The sum of the x-intercepts of lines m and n is positive.

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III


Since the slope of line m is between 0 and 1, the slope of line n must be a negative number less than -1 since line n's slope is the negative reciprocal of the slope of m. Since lines m and n intersect at a point in the first quadrant, line n (a negatively-sloped line) that passes a point in the first quadrant must have a positive x-intercept. So Roman numeral I is true.

Since the slope of m is between 0 and 1 and line m passes through point (a, b), where a > b > 0 in the first quadrant, 3 cases are possible for line m:

1) If the y-intercept is positive, then the x-intercept is negative.

2) If the y-intercept is negative, then the x-intercept is positive.

3) If the y-intercept is 0, the x-intercept is also 0.

We see that in any of these 3 cases, the product is either negative or 0. So Roman numeral II is true.

As we can see from above, the x-intercept of line n is positive, and the x-intercept of line m is negative. However, since we don’t know the values of the two x-intercepts, we can’t tell whether the sum of the x-intercepts is positive or not.

Answer: D
_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
TTP - Target Test Prep Logo
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.


Originally posted by ScottTargetTestPrep on 27 May 2019, 19:00.
Last edited by ScottTargetTestPrep on 05 Jun 2019, 06:50, edited 1 time in total.
Intern
Intern
avatar
B
Joined: 29 Apr 2019
Posts: 20
CAT Tests
Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post 03 Jun 2019, 22:46
ScottTargetTestPrep IanStewart

if you could please share your graphs here. how are we sure that y-intercept is definitely positive?
GMAT Tutor
avatar
G
Joined: 24 Jun 2008
Posts: 1662
Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

Show Tags

New post 04 Jun 2019, 09:00
1
ScottTargetTestPrep wrote:
Since the slope of m is between 0 and 1 and it passes through point (a, b), where a > b > 0 in the first quadrant, the y-intercept of line m must be positive, and the x-intercept must be negative. Therefore, the product of the x- and y-intercepts of line m is negative. So Roman numeral II is true.


This is not correct, and might account for Sourav's question above. We have no way to know if the y-intercept of line m is positive or negative. If, say, line m passes through the point (2, 1), and has a slope of 1/100, then its y-intercept is definitely positive. But if line m passes through, say, (10000000, 1), and has a slope of 1/2, its y-intercept is very negative.

All we can say is what item II says: the product of the line's two intercepts is not positive. When its y-intercept is positive, its x-intercept is negative and vice versa.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
GMAT Club Bot
Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0   [#permalink] 04 Jun 2019, 09:00
Display posts from previous: Sort by

Perpendicular lines m and n intersect at point (a, b), where a > b > 0

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


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