It is currently 24 Feb 2018, 10:09

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

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

The equation of line n is y = 4/3*x - 100. What is the small

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

Hide Tags

2 KUDOS received
Manager
Manager
avatar
Joined: 26 Mar 2007
Posts: 78
Concentration: General Management, Leadership
Schools: Thunderbird '15
The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 13 Jul 2011, 09:50
2
This post received
KUDOS
21
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

60% (01:43) correct 40% (01:35) wrong based on 543 sessions

HideShow timer Statistics

The equation of line n is y = 4/3*x - 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line n?

A. 48
B. 50
C. 60
D. 75
E. 100
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Dec 2013, 06:24, edited 1 time in total.
Renamed the topic and edited the question.
1 KUDOS received
Manager
Manager
avatar
Joined: 03 Jun 2010
Posts: 167
Location: United States (MI)
Concentration: Marketing, General Management
WE: Business Development (Consumer Products)
GMAT ToolKit User Reviews Badge
Re: Distance from the origin [#permalink]

Show Tags

New post 13 Jul 2011, 10:10
1
This post received
KUDOS
4
This post was
BOOKMARKED
kannn wrote:
The equation of line n is y = \(\frac{4}{3}\) x – 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line n ?

a) 48
b) 50
c) 60
d) 75
e) 100

The shortest line is a perpendicular, which goes through \((0;0)\)
We know that lines (\(y=kx+b\)) are perpendicular when \(k1*k2=-1\).
So, \((4/3)*k2=-1\)
k2=3/4
Line has point (0;0) also.
So, \(y=(3/4)x\)
This line intersects with \(y=\frac{4}{3}x-100\).
Let's find the intersection point.
\(\frac{4}{3}x-100=-\frac{3}{4}\)
\(x=48\)
\(y=\frac{4}{3}*48-100\)
\(y=36\)
\((48;36)\)
The shortest distance from (0;0) to above mentioned point is
\(\sqrt{48*48+36*36}\)=\(60\)
(C)
10 KUDOS received
Current Student
User avatar
Joined: 05 Jun 2013
Posts: 57
GMAT 1: 740 Q50 V40
Re: Distance from the origin [#permalink]

Show Tags

New post 07 Dec 2013, 09:13
10
This post received
KUDOS
3
This post was
BOOKMARKED
A shortcut to find the shortest distance between a point - P(m,n) and a line that doesn't contain it:

1. Re-arrange equation of the line such that its in the form ax+by+c=0
2. Put the "point in the line", i.e substitute the a and the b in the line's equation with m and n respectively.
3. The distance is = absolute value of [a(m)+b(n)+c]/[(a^2+b^2)^1/2]

Taking the example of the problem at hand,
Your point is P(0,0) and line is 4x-3y-300=0

So the distance between (0,0) and the line is: [4(0)-3(0)-300]/[(4^2+3^2)^1/2]

= 300/5 = 60 units
Intern
Intern
avatar
Status: FTW
Joined: 01 Apr 2012
Posts: 12
Location: India
Concentration: International Business, Leadership
GMAT Date: 09-27-2014
GPA: 3.5
WE: Consulting (Venture Capital)
GMAT ToolKit User
Re: Distance from the origin [#permalink]

Show Tags

New post 08 Dec 2013, 06:36
Plotting the line on the graph gives us a right triangle with, say base=75 and height =100. Now area on this triangle will be 1/2 *b*h= 1/2*75*100. Now, the shortest distance from the origin will be an altitude (say, x) drawn to the hypotenuse (whose length is 125; we can calculate because we already have the base and height) of the same triangle. Now 1/2*125*x= 1/2*75*100.
Hence, x= 60
5 KUDOS received
Current Student
User avatar
Joined: 06 Sep 2013
Posts: 1954
Concentration: Finance
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 28 Dec 2013, 06:27
5
This post received
KUDOS
1
This post was
BOOKMARKED
kannn wrote:
The equation of line n is y = 4/3*x - 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line n?

A. 48
B. 50
C. 60
D. 75
E. 100


1. Draw your line first
2. Note the pythagorean triple, that is divide by 25 to get your 3,4,5
3. Given proportions Hypothenuse is 125
4. Now area is either b*h or leg*leg (both are divided by 2 but in this case we can cancel them out)
5. 75(100) = 125 (x). 'x' represents the height which is perpendicular to the base and is the smallest distance from the origin
6. x = 60

Answer is C

Bonus method: If y'all want to save some time in there's a formula for distance from point to line that has been mentioned in the posts above but anyways this is how it would work for this problem

d = 100 / sqrt (1^2 + (4/3)^2) = 100 / 5/3 = 60

Sanity check, answer still 60

Hope this gives a hand

Hope it helps
Cheers!

J :)

Last edited by jlgdr on 31 Mar 2014, 07:58, edited 1 time in total.
SDA Bocconi Thread Master
avatar
Joined: 27 Dec 2012
Posts: 36
Location: India
Concentration: Technology, Entrepreneurship
GMAT 1: 660 Q48 V33
GMAT 2: 730 Q49 V40
WE: Engineering (Energy and Utilities)
Premium Member
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 09 Jan 2014, 13:51
distance from (0,0) to any point on given line is Sqrt( (x-0)^2 + (y-0)^2)

For minimum distance this can be differentiated and equated to zero.

Distance (D)^2 = x^2 + y^2 --------------- (1)
put y= 4x/3 -100

(1) becomes 25x^2/9 -800x/3 +10000

differentiate this equation and equate it to zero, x= 48
so y=-36.

Distance then is sqrt(48^2 + 36^2) = 60

(C)


DJ
Intern
Intern
avatar
Joined: 25 Nov 2013
Posts: 16
Concentration: Marketing, International Business
GMAT Date: 02-14-2014
GPA: 2.3
WE: Other (Internet and New Media)
Re: Distance from the origin [#permalink]

Show Tags

New post 12 Jan 2014, 09:01
reetskaur wrote:
Plotting the line on the graph gives us a right triangle with, say base=75 and height =100. Now area on this triangle will be 1/2 *b*h= 1/2*75*100. Now, the shortest distance from the origin will be an altitude (say, x) drawn to the hypotenuse (whose length is 125; we can calculate because we already have the base and height) of the same triangle. Now 1/2*125*x= 1/2*75*100.
Hence, x= 60


Can u please explain this in more detail. Thank you
1 KUDOS received
Intern
Intern
avatar
Status: Researching for Schools
Joined: 21 Apr 2013
Posts: 19
Location: United States
Concentration: Leadership, General Management
GMAT 1: 640 Q45 V34
GMAT 2: 730 Q49 V40
WE: Project Management (Computer Software)
Premium Member Reviews Badge
Re: Distance from the origin [#permalink]

Show Tags

New post 17 Feb 2014, 09:44
1
This post received
KUDOS
Vidhi1 wrote:
reetskaur wrote:
Plotting the line on the graph gives us a right triangle with, say base=75 and height =100. Now area on this triangle will be 1/2 *b*h= 1/2*75*100. Now, the shortest distance from the origin will be an altitude (say, x) drawn to the hypotenuse (whose length is 125; we can calculate because we already have the base and height) of the same triangle. Now 1/2*125*x= 1/2*75*100.
Hence, x= 60


Can u please explain this in more detail. Thank you


In the equation of the line put X=0 you would get Y=-100 and putting Y=0 would give you X=75. So the line intercepts the X and Y axis at 75 and -100 respectively. This creates a right angle triangle. We can use the Pythagoras theorem to calculate the Hypotnuse. Notice that the trip let we have here should be 3(25), 4(25), 5(25) (we have arms as 75 and 100 so the hypotnuse should be 125). Now the area of the traingle formed would be

1/2 * 75 * 100 = 1/2 * 125 * H (This H would be the closest distnace of line from the origin)

This gives H=60.

Hope this helps!!!
Manager
Manager
avatar
Joined: 17 Nov 2013
Posts: 177
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 17 Feb 2014, 15:00
Is the goal as described as the attachment.

I follow the area of the triangle is (100*75)/2, but why are you setting the area of the triangle to (125*H)/2. I dont understand the second formula logic. Please explain.
Attachments

Book1.xlsx [10.95 KiB]
Downloaded 74 times

To download please login or register as a user

Current Student
User avatar
Joined: 06 Sep 2013
Posts: 1954
Concentration: Finance
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 17 Feb 2014, 15:02
lalania1 wrote:
Is the goal as described as the attachment.

I follow the area of the triangle is (100*75)/2, but why are you setting the area of the triangle to (125*H)/2. I dont understand the second formula logic. Please explain.


Yes buddy that's correct.
Now remember that the area of a triangle is b*h/2. Therefore one can use both legs or the base and height to find the area.
So basically you can equal both areas and solve for the distance you correctly mentioned in the excel.

Hope this clarifies
Cheers
J
1 KUDOS received
Manager
Manager
avatar
Joined: 18 Oct 2013
Posts: 83
Location: India
Concentration: Technology, Finance
GMAT 1: 580 Q48 V21
GMAT 2: 530 Q49 V13
GMAT 3: 590 Q49 V21
WE: Information Technology (Computer Software)
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 23 Feb 2014, 07:06
1
This post received
KUDOS
4
This post was
BOOKMARKED
This can be solve in two steps and without any complex calculation.

Given : equation of line as y=(4/3)x -100. So the line intercept the axes at (0,-100) and (75,0).
This can be considered a right angle triangle with right angle at (0,0) . So Base=100 , Height=75 and Hypotenuse =125 (By Pythagoras triplet)

So a perpendicular from the (0,0) to hypotenuse will be the answer.

Area of triangle= 0.5*100*75=0.5*125* x
=> x=60;

SO answer is 60
Intern
Intern
avatar
Joined: 19 Feb 2014
Posts: 17
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 23 Feb 2014, 11:28
Good step by step solution jlgdr!
Retired Moderator
User avatar
Joined: 20 Dec 2013
Posts: 185
Location: United States (NY)
GMAT 1: 640 Q44 V34
GMAT 2: 710 Q48 V40
GMAT 3: 720 Q49 V40
GPA: 3.16
WE: Consulting (Venture Capital)
GMAT ToolKit User Premium Member Reviews Badge
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 23 Feb 2014, 17:59
agreed, nice approach jlgdr
_________________

MY GMAT BLOG - ADVICE - OPINIONS - ANALYSIS

Current Student
User avatar
Joined: 06 Sep 2013
Posts: 1954
Concentration: Finance
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 23 Feb 2014, 18:23
Thanks guys. Let me know if anything remains unclear ok?

Posted from my mobile device
Manager
Manager
avatar
B
Joined: 23 May 2013
Posts: 189
Location: United States
Concentration: Technology, Healthcare
Schools: Stanford '19 (M)
GMAT 1: 760 Q49 V45
GPA: 3.5
GMAT ToolKit User
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 12 Mar 2014, 14:10
I know calculus isn't something expected to be known on the GMAT, but you can also use calculus to quickly solve this problem:

\(D^2 = x^2 + ((4x/3)-100)^2\)

\(D^2 = x^2 + ((16/9)x^2 - (8/3)(100)x +100^2)\)

\((D^2)' = 2x + 2*(16/9)*x - (8/3)(100) = 0\)

to find the minimum value we then solve for x when \((D^2)' = 0\)

\(0 = 2x + 32x/9 - (8/3)*100

(8/3)*(100) = 50x/9

x = (8*100*9)/(3*50)\)


Canceling out terms from the numerator and denominator we get \(x = 48\)

This x value will give us the minimum distance. Plugging this back in, we get:

\(D^2 = 48^2 + ((4/3)(48) - 100)^2\)

\(D^2 = 48^2 + (64 - 100)^2 = 48^2 + 36^2\)

Noticing that 6 is a factor of both 48 and 36;

\(D^2 = 6^2*(8^2 + 6^2)\)

\(D^2 = 36*(100)^2\)

\(D = 6*10 = 60.\)

Answer: C

This isn't math that will be tested on the GMAT, but maybe it could still provide shortcuts in problems like these.
Intern
Intern
avatar
Joined: 05 Feb 2014
Posts: 46
Re: Distance from the origin [#permalink]

Show Tags

New post 17 Jun 2014, 05:43
ulm wrote:
kannn wrote:
The equation of line n is y = \(\frac{4}{3}\) x – 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line n ?

a) 48
b) 50
c) 60
d) 75
e) 100

The shortest line is a perpendicular, which goes through \((0;0)\)
We know that lines (\(y=kx+b\)) are perpendicular when \(k1*k2=-1\).
So, \((4/3)*k2=-1\)
k2=3/4
Line has point (0;0) also.
So, \(y=(3/4)x\)
This line intersects with \(y=\frac{4}{3}x-100\).
Let's find the intersection point.
\(\frac{4}{3}x-100=-\frac{3}{4}\)
\(x=48\)
\(y=\frac{4}{3}*48-100\)
\(y=36\)
\((48;36)\)
The shortest distance from (0;0) to above mentioned point is
\(\sqrt{48*48+36*36}\)=\(60\)
(C)


Hi , can you please explain how K2 = 3/4 , should't it be -3/4 ?
Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 43896
Re: Distance from the origin [#permalink]

Show Tags

New post 18 Jun 2014, 05:08
Expert's post
1
This post was
BOOKMARKED
gauravsoni wrote:
ulm wrote:
kannn wrote:
The equation of line n is y = \(\frac{4}{3}\) x – 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line n ?

a) 48
b) 50
c) 60
d) 75
e) 100

The shortest line is a perpendicular, which goes through \((0;0)\)
We know that lines (\(y=kx+b\)) are perpendicular when \(k1*k2=-1\).
So, \((4/3)*k2=-1\)
k2=3/4
Line has point (0;0) also.
So, \(y=(3/4)x\)
This line intersects with \(y=\frac{4}{3}x-100\).
Let's find the intersection point.
\(\frac{4}{3}x-100=-\frac{3}{4}\)
\(x=48\)
\(y=\frac{4}{3}*48-100\)
\(y=36\)
\((48;36)\)
The shortest distance from (0;0) to above mentioned point is
\(\sqrt{48*48+36*36}\)=\(60\)
(C)


Hi , can you please explain how K2 = 3/4 , should't it be -3/4 ?


For one line to be perpendicular to another, the relationship between their slopes has to be negative reciprocal \(-\frac{1}{m}\). In other words, the two lines are perpendicular if and only if the product of their slopes is \(-1\).

So, yes, you are right, the slope of a line which is perpendicular to y = 4/3*x - 100 is -3/4 (-1/(4/3) = -3/4).

Theory on Coordinate Geometry: math-coordinate-geometry-87652.html

All DS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=41
All PS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=62


Hope this helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 08 Dec 2013
Posts: 41
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 09 Nov 2014, 20:25
1
This post was
BOOKMARKED
the distance of a point(x1,y2) from a line Ax+by+c=0 is given by..
Modulus{(Ax1+By2+c)/sqrt(A^2+B^2)}

high school formula, I guess.
Current Student
avatar
Joined: 18 Apr 2015
Posts: 5
Location: United States
Concentration: Operations, General Management
GMAT 1: 710 Q50 V35
GPA: 3.12
WE: Manufacturing and Production (Manufacturing)
The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 10 May 2015, 00:23
There is a formula to solve such questions.
The min distance between points (p,q) and the line segment ax+by+c=0 is given by
(|pa+bq+c|)/[sqrt(a^2+b^2)]




- Rajat
Expert Post
EMPOWERgmat Instructor
User avatar
D
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 11070
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: The equation of line n is y = 4/3*x - 100. What is the small [#permalink]

Show Tags

New post 10 May 2015, 14:57
Hi All,

This question is LOADED with pattern-matching shortcuts. If you can spot these shortcuts, then you can save LOTS of time and avoid much of the math that other Test Takers would need to do to answer this question.

The shortcuts are:
[Reveal] Spoiler:
1) Draw a quick graph of the line; you should notice that you have a 3/4/5 right triangle with sides 75/100/125
2) You can 'cut' this big triangle into 2 smaller right triangles that are ALSO 3/4/5 right triangles
3) Using the 75 and 100 as reference, you can fill in the missing sides and end up with a 45/60/75 triangle and a 60/80/100 triangle
4) The common side also happens to be the shortest length = 60


GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free
  Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Re: The equation of line n is y = 4/3*x - 100. What is the small   [#permalink] 10 May 2015, 14:57

Go to page    1   2    Next  [ 25 posts ] 

Display posts from previous: Sort by

The equation of line n is y = 4/3*x - 100. What is the small

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


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.