Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 27 May 2016, 13:48

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# What is the least possible distance between a point on the

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [8] , given: 9856

What is the least possible distance between a point on the [#permalink]

### Show Tags

11 Oct 2009, 18:19
8
KUDOS
Expert's post
15
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

56% (02:56) correct 44% (02:35) wrong based on 385 sessions

### HideShow timer Statistics

What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?

A. 1.4

B. $$\sqrt{2}$$

C. 1.7

D. $$\sqrt{3}$$

E. 2.0
[Reveal] Spoiler: OA

_________________
Manager
Joined: 02 Jan 2009
Posts: 96
Location: India
Schools: LBS
Followers: 2

Kudos [?]: 69 [2] , given: 6

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

12 Oct 2009, 00:54
2
KUDOS
Bunuel wrote:
What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?

A) 1.4
B) sqrt (2)
C) 1.7
D) sqrt (3)
E) 2.0

Good problem again. Takes into account a lot of co ordinate geometry fundas.

eqn of circle = x^2 + y^2 = 1, center = (0,0) radius = 1

min dist of line from circle = dist of line from the center - radius

Make the distance of the line from the circle to be 0 and we see that it becomes a tangent to the circle.

Now we know if we draw a line from the center to the point where the tangent touches the circle the line and the tangent are perpendicular to each other.

So we need to find the equation of this line first.

We can take the line back where it was now

Since the lines are perpendicular m1 x m2 = -1

m of line = 3/4

so slope of the new line = -4/3

Since the line passes through the origin (center of circle) its eqn => y=-4/3x

now we need to get the point of intersection of our two lines, which comes out to be (36/25,-48/25)

now get the distance of this point from the origin and subtract the radius from it.

Comes to 1.4 (may have made calculation errors )

So A.

Comes under 2 mins.

Bunuel, great work with the Questions. I suggest you make a single thread and keep updating it. People can subscribe to that and also it will help new guys read all the probs and solutions in one thread.
_________________

The Legion dies, it does not surrender.

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2797
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 217

Kudos [?]: 1415 [10] , given: 235

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

12 Oct 2009, 06:45
10
KUDOS
Lets do in 1 min.

Use the formula

D = | Am+Bn+C|/ SQRT(A^2 + B^2) where Ax+By+C = 0

put (m,n) =0,0 = center of circle

we get D = 12/5 thus required distance is D-1 = 12/5 -1 = 7/5 = 1.4

we don't require points
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [1] , given: 9856

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

12 Oct 2009, 09:28
1
KUDOS
Expert's post
Set of Tough & tricky questions is now combined in one thread: tough-tricky-set-of-problms-85211.html

You can continue discussions and see the solutions there.

_________________
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2797
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 217

Kudos [?]: 1415 [1] , given: 235

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

14 Oct 2009, 05:23
1
KUDOS
I have learnt this in my school time that distance from one point to a line is what stated above.

Actually its same like finding the line which originates from the distant point and then intersect at the required line,
and then we find the point of intersection and then using the formula we calculate distance between the two points.

But even if u dont rem the formula what u can do is....

suppose line is ax+by-c=0 now when y=0 , x=c/a and when x=0 , y = c/b

area of triangle formed by these 2 points and center (0,0) is 1/2 * c/a * c/b = c^2/2ab

now this is equal to 1/2 * D1 * D2 , where D1 is distance between points on x and y coordinates of the line which is sqrt [ (c/a)^2 + (b/a) ^2 ] and D2 is the required perpendicular distance on the line.

Equate them and u will get the ans. Its very easy concept and I m not that good in explaining here I think sry for that.

If you still dont get this please letme know I will explain again.
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Manager
Joined: 24 Aug 2009
Posts: 150
Followers: 5

Kudos [?]: 78 [0], given: 46

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

14 Dec 2009, 20:55
gurpreetsingh wrote:
Lets do in 1 min.

Use the formula

D = | Am+Bn+C|/ SQRT(A^2 + B^2) where Ax+By+C = 0

put (m,n) =0,0 = center of circle

we get D = 12/5 thus required distance is D-1 = 12/5 -1 = 7/5 = 1.4

we don't require points

I am too weak in this section, could you please explain...what is this formula and why did you do D-1?
Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [4] , given: 9856

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

15 Dec 2009, 06:58
4
KUDOS
Expert's post
4
This post was
BOOKMARKED
ISBtarget wrote:
I am too weak in this section, could you please explain...what is this formula and why did you do D-1?

First note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).

Now we can do this by finding the equation of a line perpendicular to given line $$y=\frac{3}{4}*x-3$$ (we know it should cross origin and cross given line, so we can write the formula of it), then find the croos point of these lines and then the distance between the origin and this point. But it's very lengthy way.

There is another, shorter one. Though I've never seen any GMAT question requiring the formula used in it.

We know the formula to calculate the distance between two points $$(x1,y1)$$ and $$(x2,y2)$$: $$d=\sqrt{(x1-x2)^2+(y1-y2)^2}$$ BUT there is a formula to calculate the distance between the point (in our case origin) and the line:

DISTANCE BETWEEN THE LINE AND POINT:
Line: $$ay+bx+c=0$$, point $$(x1,y1)$$

$$d=\frac{|ay1+bx1+c|}{\sqrt{a^2+b^2}}$$

DISTANCE BETWEEN THE LINE AND ORIGIN:
As origin is $$(0,0)$$ -->

$$d=\frac{|c|}{\sqrt{a^2+b^2}}$$

So in our case it would be: $$d=\frac{|-3|}{\sqrt{1^2+(\frac{3}{4})^2}}=\frac{12}{5}=2.4$$

So the shortest distance would be: $$2.4-1(radius)=1.4$$

P.S. Also note that when we have $$x^2+y^2=k$$, we have circle (as we have $$x^2$$ and $$y^2$$), it's centered at the origin (as coefficients of $$x$$ and $$y$$ are $$1$$) and the radius of that circle $$r=\sqrt{k}$$.

You can check the link of Coordinate Geometry below for more.
_________________
Manager
Joined: 24 Aug 2009
Posts: 150
Followers: 5

Kudos [?]: 78 [0], given: 46

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

15 Dec 2009, 13:50
Bunuel wrote:
ISBtarget wrote:
I am too weak in this section, could you please explain...what is this formula and why did you do D-1?

First note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).

Now we can do this by finding the equation of a line perpendicular to given line $$y=\frac{3}{4}*x-3$$ (we know it should cross origin and cross given line, so we can write the formula of it), then find the croos point of these lines and then the distance between the origin and this point. But it's very lengthy way.

There is another, shorter one. Though I've never seen any GMAT question requiring the formula used in it.

We know the formula to calculate the distance between two points $$(x1,y1)$$ and $$(x2,y2)$$: $$d=\sqrt{(x1-x2)^2+(y1-y2)^2}$$ BUT there is a formula to calculate the distance between the point (in our case origin) and the line:

DISTANCE BETWEEN THE LINE AND POINT:
Line: $$ay+bx+c=0$$, point $$(x1,y1)$$

$$d=\frac{|ay1+bx1+c|}{\sqrt{a^2+b^2}}$$

DISTANCE BETWEEN THE LINE AND ORIGIN:
As origin is $$(0,0)$$ -->

$$d=\frac{|c|}{\sqrt{a^2+b^2}}$$

So in our case it would be: $$d=\frac{|-3|}{\sqrt{1^2+(\frac{3}{4})^2}}=\frac{12}{5}=2.4$$

So the shortest distance would be: $$2.4-1(radius)=1.4$$

P.S. Also note that when we have $$x^2+y^2=k$$, we have circle (as we have $$x^2$$ and $$y^2$$), it's centered at the origin (as coefficients of $$x$$ and $$y$$ are $$1$$) and the radius of that circle $$r=\sqrt{k}$$.

You can check the link of Coordinate Geometry below for more.

Awesome man, why wouldnt you start a quant training program....excellent
Manager
Joined: 24 Aug 2009
Posts: 150
Followers: 5

Kudos [?]: 78 [0], given: 46

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

17 Dec 2009, 13:55
Hi

One more question...sounds silly but can you help

why are you doing D-1 , you are calculating the distance between a point on the circle and the line , 0,0 is a point on the circle, why cant 2.4 be the answer
Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [0], given: 9856

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

17 Dec 2009, 14:11
Expert's post
2
This post was
BOOKMARKED
Minimum distance from the circle to the line would be:
Length of perpendicular from the origin to the line (as the circle is centered at the origin) - The radius of a circle (which is 1).

(0,0) is not the point on the circle, it's the center of the circle with radius 1 (circle is centered at the origin).

2.4 is the distance from the line to the origin (the center of the circle), so we should subtract the length of the radius to get the distance from the line to the circle.

ANOTHER SOLUTION:

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
$$(x-a)^2+(y-b)^2=r^2$$

If the circle is centered at the origin (0, 0), then the equation simplifies to:
$$x^2+y^2=r^2$$

So, the circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$.

Then note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).

So we should find the length of perpendicular, or the height of the right triangle formed by the X and Y axis and the line $$y = \frac{{3}}{{4}}x-3$$.

The legs would be the value of x for y=0 (x intercept) --> y=0, x=4 --> $$leg_1=4$$.
and the value of y for x=0 (y intercept) --> x=0, y=-3 --> $$leg_2=3$$.

So we have the right triangle with legs 4 and 3 and hypotenuse 5. What is the height of this triangle (perpendicular from right angle to the hypotenuse)? As perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle: $$\frac{height}{leg_1}=\frac{leg_2}{hypotenuse}$$ --> $$\frac{height}{3}=\frac{4}{5}$$ --> $$height=2.4$$.

$$Distance=height-radius=2.4-1=1.4$$

Hope it's helps.
_________________
Manager
Joined: 23 Oct 2009
Posts: 72
Followers: 1

Kudos [?]: 25 [0], given: 14

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

31 Dec 2010, 08:45
Have questions based on this concept (i.e. Circle on a plane) ever been tested by the GMAT?
_________________

Consider giving Kudos if you liked my post. Thanks!

Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [1] , given: 9856

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

31 Dec 2010, 09:00
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
rahul321 wrote:
Have questions based on this concept (i.e. Circle on a plane) ever been tested by the GMAT?

Yes. Check the following GMAT prep questions:
point-on-a-circle-106016.html
does-line-k-touch-circle-or-not-101471.html
plane-geometry-semicircle-from-gmatprep-85154.html
does-the-point-p-touch-the-circle-101485.html
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6578
Location: Pune, India
Followers: 1791

Kudos [?]: 10774 [14] , given: 211

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

31 Dec 2010, 10:17
14
KUDOS
Expert's post
4
This post was
BOOKMARKED
A rather nice question and I think Bunuel has already explained sufficient alternative solutions. I just want to add a method of finding the altitude of a right triangle here that I think is particularly neat (discussed in Veritas Geometry book)
Attachment:

Ques2.jpg [ 8.1 KiB | Viewed 16757 times ]

I want to find x here since (x - 1) will be the minimum distance from the circle to the line (as explained above)

Area of the given right triangle = (1/2)*3*4 (3 is base and 4 is altitude)= (1/2)*5*x (5 is base and x is altitude)
So x = 2.4

Finding the area of the original triangle in two different ways and equating it will help you find the altitude.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 33052
Followers: 5765

Kudos [?]: 70652 [0], given: 9856

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

31 Dec 2010, 10:23
Expert's post
1
This post was
BOOKMARKED
VeritasPrepKarishma wrote:
A rather nice question and I think Bunuel has already explained sufficient alternative solutions. I just want to add a method of finding the altitude of a right triangle here that I think is particularly neat (discussed in Veritas Geometry book)
Attachment:
Ques2.jpg

I want to find x here since (x - 1) will be the minimum distance from the circle to the line (as explained above)

Area of the given right triangle = (1/2)*3*4 (3 is base and 4 is altitude)= (1/2)*5*x (5 is base and x is altitude)
So x = 2.4

Finding the area of the original triangle in two different ways and equating it will help you find the altitude.

_________________
Manager
Joined: 23 Oct 2009
Posts: 72
Followers: 1

Kudos [?]: 25 [0], given: 14

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

01 Jan 2011, 03:08
Bunuel wrote:
rahul321 wrote:
Have questions based on this concept (i.e. Circle on a plane) ever been tested by the GMAT?

Yes. Check the following GMAT prep questions:
point-on-a-circle-106016.html
does-line-k-touch-circle-or-not-101471.html
plane-geometry-semicircle-from-gmatprep-85154.html
does-the-point-p-touch-the-circle-101485.html

Thanks. I'm just reading the GMATclub Coordinate Geometry post now...Great stuff!!

It's surprising that the OG Quant book doesn't remotely touch upon the 'Circle on a plane' topic even though it is tested!
_________________

Consider giving Kudos if you liked my post. Thanks!

Manager
Joined: 18 Jun 2010
Posts: 148
Followers: 0

Kudos [?]: 30 [1] , given: 2

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

22 Oct 2011, 11:35
1
KUDOS
Bunuel wrote:
ISBtarget wrote:
I am too weak in this section, could you please explain...what is this formula and why did you do D-1?

First note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).

Now we can do this by finding the equation of a line perpendicular to given line $$y=\frac{3}{4}*x-3$$ (we know it should cross origin and cross given line, so we can write the formula of it), then find the croos point of these lines and then the distance between the origin and this point. But it's very lengthy way.

There is another, shorter one. Though I've never seen any GMAT question requiring the formula used in it.

We know the formula to calculate the distance between two points $$(x1,y1)$$ and $$(x2,y2)$$: $$d=\sqrt{(x1-x2)^2+(y1-y2)^2}$$ BUT there is a formula to calculate the distance between the point (in our case origin) and the line:

DISTANCE BETWEEN THE LINE AND POINT:
Line: $$ay+bx+c=0$$, point $$(x1,y1)$$

$$d=\frac{|ay1+bx1+c|}{\sqrt{a^2+b^2}}$$

DISTANCE BETWEEN THE LINE AND ORIGIN:
As origin is $$(0,0)$$ -->

$$d=\frac{|c|}{\sqrt{a^2+b^2}}$$

So in our case it would be: $$d=\frac{|-3|}{\sqrt{1^2+(\frac{3}{4})^2}}=\frac{12}{5}=2.4$$

So the shortest distance would be: $$2.4-1(radius)=1.4$$

P.S. Also note that when we have $$x^2+y^2=k$$, we have circle (as we have $$x^2$$ and $$y^2$$), it's centered at the origin (as coefficients of $$x$$ and $$y$$ are $$1$$) and the radius of that circle $$r=\sqrt{k}$$.

You can check the link of Coordinate Geometry below for more.

Brilliant explanation.
Intern
Joined: 23 Sep 2008
Posts: 24
Followers: 0

Kudos [?]: 24 [0], given: 137

Re: Tough and tricky 7: distance between the circle an line [#permalink]

### Show Tags

06 Dec 2012, 12:45
Bunuel wrote:
Minimum distance from the circle to the line would be:
Length of perpendicular from the origin to the line (as the circle is centered at the origin) - The radius of a circle (which is 1).

(0,0) is not the point on the circle, it's the center of the circle with radius 1 (circle is centered at the origin).

2.4 is the distance from the line to the origin (the center of the circle), so we should subtract the length of the radius to get the distance from the line to the circle.

ANOTHER SOLUTION:

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
$$(x-a)^2+(y-b)^2=r^2$$

If the circle is centered at the origin (0, 0), then the equation simplifies to:
$$x^2+y^2=r^2$$

So, the circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$.

Then note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).

So we should find the length of perpendicular, or the height of the right triangle formed by the X and Y axis and the line $$y = \frac{{3}}{{4}}x-3$$.

The legs would be the value of x for y=0 (x intercept) --> y=0, x=4 --> $$leg_1=4$$.
and the value of y for x=0 (y intercept) --> x=0, y=-3 --> $$leg_2=3$$.

So we have the right triangle with legs 4 and 3 and hypotenuse 5. What is the height of this triangle (perpendicular from right angle to the hypotenuse)? As perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle: $$\frac{height}{leg_1}=\frac{leg_2}{hypotenuse}$$ --> $$\frac{height}{3}=\frac{4}{5}$$ --> $$height=2.4$$.

$$Distance=height-radius=2.4-1=1.4$$

Hope it's helps.

Hi Bunuel,

I could not understand the last but one step.

how did you take the ratios?
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 9645
Followers: 465

Kudos [?]: 120 [0], given: 0

Re: What is the least possible distance between a point on the [#permalink]

### Show Tags

07 Feb 2014, 22:15
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.
_________________
Manager
Joined: 25 Oct 2013
Posts: 173
Followers: 1

Kudos [?]: 42 [0], given: 56

Re: What is the least possible distance between a point on the [#permalink]

### Show Tags

09 Feb 2014, 07:50
Took quite some time(6+ mins) to solve . But I took the same approach as explained by VeritasPrepKarishma. I just hope these ideas click on time. Nice question.
_________________

Click on Kudos if you liked the post!

Practice makes Perfect.

Intern
Joined: 22 Mar 2014
Posts: 32
Followers: 0

Kudos [?]: 43 [1] , given: 21

Re: What is the least possible distance between a point on the [#permalink]

### Show Tags

10 Jul 2014, 08:19
1
KUDOS
1
This post was
BOOKMARKED
For the people still troubled with this questions -

First of all this questions tests relatively advanced skills in mathematics and hence IMO can not be a part of the real GMAT.

However, to find the answer of this Q we need to follow below mentioned steps:

1) find the distance of the line from origin : this distance should be the shortest possible distance
2) as all points on circle are equi distance from the origin, we need to find the shortest distance of line from origin and subtract radius from it to get our answer
3) to get shortest distance we need to actually find length of perpendicular line which starts from origin and ends at our given line

3-a) One of the method to solve for (3) is using the equation - |ax1 + by1 + c| /sqrt (a^2+b^2) (this formula you need to remember) - read posts by bunuel or gurpreet for more details.
3-b) Another method is to find the equation of perpendicular line and then find an intersection point of this perpendicular line with our given line. Now find the distance between this point to origin (PHEW) - I surely can't do all this in less 2 minutes and be accurate to the second decimal point [ remember our options are 1.4 and sqrt (2) = 1.41 ]

Finally, for the purpose of GMAT only, i would advise you should not be worried if you can't remember this formula or find this question too difficult.
_________________

-Sameer
Press Kudos if the post helped

Re: What is the least possible distance between a point on the   [#permalink] 10 Jul 2014, 08:19

Go to page    1   2    Next  [ 22 posts ]

Similar topics Replies Last post
Similar
Topics:
2 In the coordinate plane, what is the distance between points (3, 5) an 5 21 Feb 2016, 10:12
What is the distance between point A and point C? 3 12 May 2015, 12:48
7 What is the greatest distance between two points in a 4 30 Oct 2012, 09:09
9 What is the least possible distance between a point on the c 8 21 Mar 2011, 21:40
42 What is the least possible distance between a point on the 21 12 Apr 2010, 01:26
Display posts from previous: Sort by