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M13 Q12

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M13 Q12 [#permalink] New post 04 Dec 2011, 18:56
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What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = \frac{3}{4}x - 3 ?

1.4
\sqrt{2}
1.7
\sqrt{3}
2.0

Can anyone Please explain this Question??
[Reveal] Spoiler: OA
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Re: M13 Q12 [#permalink] New post 04 Dec 2011, 23:45
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CracktheGmat2010 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 = \frac{3}{4}x - 3 ?

1.4
\sqrt{2}
1.7
\sqrt{3}
2.0

Can anyone Please explain this Question??


The question has been discussed before. This is my take on it.

Look at the diagram below and forget the circle for the time being. What is the minimum distance from the center (0,0) to the line? It will the perpendicular from the center to the line, right? (shown by the bold line)

Attachment:
File.jpg
File.jpg [ 17.76 KiB | Viewed 984 times ]


Now think, what will be the shortest distance from the circle to the line? It will be 1 unit less than the distance from the center to the line. Can we say it will be the least in case of the bold line which is perpendicular to the given line? Yes, it will be because in all other cases, the lines are longer than the perpendicular and hence (line - 1) will also be longer.

Then, let's try to find the length of the bold line, x.
Since hypotenuse is 5,
(1/2)*3*4 = (1/2)*5*x = Area of triangle made by the co-ordinate axis and the given line
x = 2.4
So minimum distance is 2.4 - 1 = 1.4
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Re: M13 Q12 [#permalink] New post 07 Dec 2011, 07:15
VeritasPrepKarishma wrote:
CracktheGmat2010 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 = \frac{3}{4}x - 3 ?

1.4
\sqrt{2}
1.7
\sqrt{3}
2.0

Can anyone Please explain this Question??


The question has been discussed before. This is my take on it.

Look at the diagram below and forget the circle for the time being. What is the minimum distance from the center (0,0) to the line? It will the perpendicular from the center to the line, right? (shown by the bold line)

Attachment:
File.jpg


Now think, what will be the shortest distance from the circle to the line? It will be 1 unit less than the distance from the center to the line. Can we say it will be the least in case of the bold line which is perpendicular to the given line? Yes, it will be because in all other cases, the lines are longer than the perpendicular and hence (line - 1) will also be longer.

Then, let's try to find the length of the bold line, x.
Since hypotenuse is 5,
(1/2)*3*4 = (1/2)*5*x = Area of triangle made by the co-ordinate axis and the given line
x = 2.4
So minimum distance is 2.4 - 1 = 1.4



i have one serious doubt
if we find the perpendicular distance (d) given by the formula
|ax+by+c|/(a^2+b^2)^1/2

the value comes to be 4 against your value which comes out to be 2.4
tell me where i am going wrong. :(
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Re: M13 Q12 [#permalink] New post 07 Dec 2011, 07:17
sorry i did some calculation mistake ... i am also getting 2.4 :P
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Re: M13 Q12 [#permalink] New post 12 Dec 2011, 23:15
what is the difficulty level of this question?
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Re: M13 Q12 [#permalink] New post 13 Dec 2011, 02:28
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sandeeepsharma wrote:
what is the difficulty level of this question?


700+, actually closer to 750.
It takes a degree of imagination to figure out the solution quickly.
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Re: M13 Q12 [#permalink] New post 13 Dec 2011, 03:37
thanks karishma..
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Re: M13 Q12 [#permalink] New post 13 Dec 2011, 10:00
That's an awesome question.
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Re: M13 Q12   [#permalink] 13 Dec 2011, 10:00
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