Shortest Distance between a Circle and a Line

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ManuGMAT: Junior | Next Rank: 30 Posts; Posts: 13; Joined: Thu Nov 06, 2008 12:14 pm; Thanked: 1 times

by ManuGMAT » Mon Jun 29, 2009 6:51 pm

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

A. 1.4
B. 2^1/2
C. 1.7
D 3^1/2
E. 2.0

Please explain the answer.

OA is A

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Source: Beat The GMAT — Problem Solving | Mon Jun 29, 2009 6:51 pm

ssmiles08: Master | Next Rank: 500 Posts; Posts: 472; Joined: Sun Mar 29, 2009 6:54 pm; Thanked: 56 times

by ssmiles08 » Mon Jun 29, 2009 7:53 pm

I sort of got the answer...but my method was somewhat crude...maybe some one has a better answer for you.

First I figured out the x and y intercepts of the line. (4,0) and (-3,0). this told me right away that there is some sort of a 3-4-5 triangle involved with it.

you know the radius of the circle is 1 with origin at (0,0).

when you draw it out in paper, you would have a right triangle in the 4th quadrant with 1/4 of the circle coming in the way.

The shortest distance would be the line perpendicular to the line Y=3/4X-3 and the circle.

so what I did was draw out a rectangle at the 4th quadrant (basically two 3-4-5 triangles) (0,0)(4,0)(-3,0)(4,-3)

Then I drew two diagonals with the length of 5 within the rectangle. The distance from the origin(0,0) to the midpoint of the rectangle is approximately 2.5.

the radius is 1, so you subtract 2.5 from 1 to get ~1.5 which is closest to A.

Like I said, my method was crude and I am not even sure if it right...maybe someone has a better explanation.

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Claret: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Sat Jun 06, 2009 10:48 pm; Thanked: 4 times

by Claret » Mon Jun 29, 2009 7:54 pm

perpendicular Distance(shortest distance) of a point (x1,y1) from a line ax+by+c = 0 is given by
d = |(a*x1)+(b*y1)+c|/(a^2+b^2)^1/2

Let us calculate the distance of the line from the center of the circle (0,0)
a = -3/4
b=1
c=3
x1=0
y1=0

therefore d= 12/5

distance from a point on the circle would be 12/5-1 = 1.4

hope this helps!!

Last edited by Claret on Mon Jun 29, 2009 8:08 pm, edited 1 time in total.

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Claret: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Sat Jun 06, 2009 10:48 pm; Thanked: 4 times

by Claret » Mon Jun 29, 2009 8:06 pm

ssmiles08 wrote:
the radius is 1, so you subtract 2.5 from 1 to get ~1.5 which is closest to A.

one thing to note here in this question is two answer choices are pretty close

A) 1.4 and B) (2) ^1/2 ~ 1.414

if we bring down the calculation to ~1.5 , either A or B could be the answer..

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ssmiles08: Master | Next Rank: 500 Posts; Posts: 472; Joined: Sun Mar 29, 2009 6:54 pm; Thanked: 56 times

by ssmiles08 » Tue Jun 30, 2009 7:52 am

Claret wrote:
ssmiles08 wrote:
the radius is 1, so you subtract 2.5 from 1 to get ~1.5 which is closest to A.

one thing to note here in this question is two answer choices are pretty close

A) 1.4 and B) (2) ^1/2 ~ 1.414

if we bring down the calculation to ~1.5 , either A or B could be the answer..

Yes you are right. Thank you.

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boysangur: Junior | Next Rank: 30 Posts; Posts: 19; Joined: Sat Aug 14, 2010 2:57 pm

by boysangur » Tue Sep 07, 2010 9:45 pm

O_O ...this is a possible GMAT problem?

I don't even know where to start. What does "circle x^2 + y^2 = 1" even mean??

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RCV: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Fri Jul 30, 2010 4:55 am; Location: India

by RCV » Tue Sep 07, 2010 11:06 pm

sanju, Ian, Kowinsky, please comment, found it very difficult for gmat

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sanjayviti: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Wed Jan 06, 2010 10:00 pm

by sanjayviti » Wed Sep 08, 2010 3:19 am

The answer is indeed 1.4 but I think it needs to be solved differently.

I think everybody has figured out the vertices of traingle OAB that is O (0,0), A(0,-3) and B(4,0). Now draw a perpendicular line from O to AB (Sorry I have not drawn fig). Say it touches OB at D.

Now you can find out OD= OB*Sin <OAB

= 4*3/5
= 12/5

We know that a tangent on any point on the circle is perpendicular to the radius.

Therefore, the shortest distance from any point on circle to the line AB= OD- Radius of the circle

i.e. 12/5-1= 1.4

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