February 17, 2019 February 17, 2019 07:00 AM PST 09:00 AM PST Attend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT. February 18, 2019 February 18, 2019 10:00 PM PST 11:00 PM PST We don’t care what your relationship status this year  we love you just the way you are. AND we want you to crush the GMAT!
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 26 Mar 2007
Posts: 63
Concentration: General Management, Leadership

The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
Updated on: 08 Dec 2013, 06:24
Question Stats:
59% (02:23) correct 41% (02:26) wrong based on 628 sessions
HideShow timer Statistics
The equation of line n is y = 4/3*x  100. What is the smallest possible distance in the xyplane from the point with coordinates (0, 0) to any point on line n? A. 48 B. 50 C. 60 D. 75 E. 100
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by kannn on 13 Jul 2011, 09:50.
Last edited by Bunuel on 08 Dec 2013, 06:24, edited 1 time in total.
Renamed the topic and edited the question.




Manager
Joined: 05 Jun 2013
Posts: 57

Re: Distance from the origin
[#permalink]
Show Tags
07 Dec 2013, 09:13
A shortcut to find the shortest distance between a point  P(m,n) and a line that doesn't contain it:
1. Rearrange 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 4x3y300=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




Manager
Joined: 03 Jun 2010
Posts: 143
Location: United States (MI)
Concentration: Marketing, General Management
WE: Business Development (Consumer Products)

Re: Distance from the origin
[#permalink]
Show Tags
13 Jul 2011, 10:10
kannn wrote: The equation of line n is y = \(\frac{4}{3}\) x – 100. What is the smallest possible distance in the xyplane 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 \(y=\frac{4}{3}x100\). Let's find the intersection point. \(\frac{4}{3}x100=\frac{3}{4}\) \(x=48\) \(y=\frac{4}{3}*48100\) \(y=36\) \((48;36)\) The shortest distance from (0;0) to above mentioned point is \(\sqrt{48*48+36*36}\)=\(60\) (C)



Intern
Status: FTW
Joined: 01 Apr 2012
Posts: 9
Location: India
Concentration: International Business, Leadership
GMAT Date: 09272014
GPA: 3.5
WE: Consulting (Venture Capital)

Re: Distance from the origin
[#permalink]
Show Tags
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



SVP
Joined: 06 Sep 2013
Posts: 1696
Concentration: Finance

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
Updated on: 31 Mar 2014, 07:58
kannn wrote: The equation of line n is y = 4/3*x  100. What is the smallest possible distance in the xyplane 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
Originally posted by jlgdr on 28 Dec 2013, 06:27.
Last edited by jlgdr on 31 Mar 2014, 07:58, edited 1 time in total.



SDA Bocconi Thread Master
Joined: 27 Dec 2012
Posts: 32
Location: India
Concentration: Technology, Entrepreneurship
GMAT 1: 660 Q48 V33 GMAT 2: 730 Q49 V40
WE: Engineering (Energy and Utilities)

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
09 Jan 2014, 13:51
distance from (0,0) to any point on given line is Sqrt( (x0)^2 + (y0)^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
Joined: 25 Nov 2013
Posts: 14
Concentration: Marketing, International Business
GMAT Date: 02142014
GPA: 2.3
WE: Other (Internet and New Media)

Re: Distance from the origin
[#permalink]
Show Tags
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



Intern
Status: Researching for Schools
Joined: 21 Apr 2013
Posts: 18
Location: United States
Concentration: Leadership, General Management
GMAT 1: 640 Q45 V34 GMAT 2: 730 Q49 V40
WE: Project Management (Computer Software)

Re: Distance from the origin
[#permalink]
Show Tags
17 Feb 2014, 09:44
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
Joined: 17 Nov 2013
Posts: 87

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
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 95 times



SVP
Joined: 06 Sep 2013
Posts: 1696
Concentration: Finance

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
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



Manager
Joined: 18 Oct 2013
Posts: 70
Location: India
Concentration: Technology, Finance
Schools: Duke '16, Johnson '16, Kelley '16, Tepper '16, Marshall '16, McDonough '16, Insead '14, HKUST '16, HSG '15, Schulich '15, Erasmus '16, IE April'15, Neeley '15
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
23 Feb 2014, 07:06
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
Joined: 19 Feb 2014
Posts: 17

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
23 Feb 2014, 11:28
Good step by step solution jlgdr!



Retired Moderator
Joined: 20 Dec 2013
Posts: 171
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)

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
23 Feb 2014, 17:59
agreed, nice approach jlgdr
_________________
MY GMAT BLOG  ADVICE  OPINIONS  ANALYSIS



SVP
Joined: 06 Sep 2013
Posts: 1696
Concentration: Finance

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
23 Feb 2014, 18:23
Thanks guys. Let me know if anything remains unclear ok?
Posted from my mobile device



Current Student
Joined: 23 May 2013
Posts: 186
Location: United States
Concentration: Technology, Healthcare
GPA: 3.5

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
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
Joined: 05 Feb 2014
Posts: 42

Re: Distance from the origin
[#permalink]
Show Tags
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 xyplane 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 \(y=\frac{4}{3}x100\). Let's find the intersection point. \(\frac{4}{3}x100=\frac{3}{4}\) \(x=48\) \(y=\frac{4}{3}*48100\) \(y=36\) \((48;36)\) 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 ?



Math Expert
Joined: 02 Sep 2009
Posts: 52907

Re: Distance from the origin
[#permalink]
Show Tags
18 Jun 2014, 05:08
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 xyplane 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 \(y=\frac{4}{3}x100\). Let's find the intersection point. \(\frac{4}{3}x100=\frac{3}{4}\) \(x=48\) \(y=\frac{4}{3}*48100\) \(y=36\) \((48;36)\) 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). 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? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 08 Dec 2013
Posts: 32

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
09 Nov 2014, 20:25
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.



Intern
Joined: 18 Apr 2015
Posts: 5
Location: United States
Concentration: Operations, General Management
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
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



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13546
Location: United States (CA)

Re: The equation of line n is y = 4/3*x  100. What is the small
[#permalink]
Show Tags
10 May 2015, 14:57
Hi All, This question is LOADED with patternmatching 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: 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
CoFounder & 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
[ 24 posts ]



