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Distance to Line/Point on Coordinate Plane- please help!

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Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 13 Oct 2010, 16:45
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.

Can someone help me understand this???

Thanks!


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Re: Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 13 Oct 2010, 23:04
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jjewkes wrote:
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.

Can someone help me understand this???

Thanks!


Jeremiah


The general equation of any line is ax+by+c=0
and length of perpendicular(shortest distance) from the point (m,n) to the line ax+by+c = 0 is given by

p=(modulus of (am+bn+c))/(square root of (a^2 + b^2))

If equation of line is in any other form you just convert it to form ax+by+c=0 and then apply this formula.


The second problem if i am understanding right....
You find out the length of perpendicular from origin(0,0) to the given line with the same above formula and subtract the diameter of the circle you will get the distance of circle from the line.


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Re: Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 13 Oct 2010, 23:36
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jjewkes wrote:
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.

Can someone help me understand this???

Thanks!


Jeremiah


You might find this handy if you havent read it yet : math-coordinate-geometry-87652.html

The distance formula and other geometric problems are explained well in that chapter.

The distance between point (x_0,y_0) and line ax+by+c=0 is given by :
\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}

When it comes to how far a circle is from a line, it depends really what line you are talking about, the best way to tackle these questions is to plot it out on a piece of paper, after which in most GMAT questions, the answer will be quite straight forward and you'll be able to get to it without any formulae
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Re: Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 14 Oct 2010, 02:54
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jjewkes wrote:
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.

Can someone help me understand this???

Thanks!


Jeremiah


Please note that it's highly unlikely that you'll need this for GMAT.

DISTANCE BETWEEN THE LINE AND POINT:
Line: ay+bx+c=0, point (x_1,y_1)

d=\frac{|ay_1+bx_1+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}}

Question about "THE DISTANCE BETWEEN THE CIRCLE AND THE LINE": tough-tricky-set-of-problms-85211.html#p638361 There are two solutions for this problem, one uses the formula above and another right triangle properties and basics of coordinate geometry.

Hope it helps.
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Re: Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 14 Oct 2010, 10:38
jjewkes wrote:
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.


I'd point out that while I've seen the occasional prep company question that asks about distances from points to lines, I've never, in ten years in this field, seen a real GMAT question that does. So while memorizing formulas like the one in the posts above might help you on a high school coordinate geometry test, it is extremely unlikely to make any difference on the GMAT.

And if you were given some point (a, b) and some line y = mx + c, and asked to find the perpendicular distance from (a,b) to the line, you could always:

* find the equation of the line perpendicular to y = mx + c which contains (a,b). That line has a slope of -1/m, and you can find its y-intercept by plugging in the point

* now find where the first line and its perpendicular intersect by solving the two equations together - that will be at some point (d,e)

* find the distance from (d,e) to (a,b) to get the answer

Following those steps, you can derive the formula in the posts above. You may well need to do one of the three things above in a question, but you're very unlikely to need to do all three, so I think it's vastly preferable to learn each concept individually rather than memorizing formulas that you'll most likely never have occasion to use.
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Re: Distance to Line/Point on Coordinate Plane- please help! [#permalink] New post 15 Oct 2010, 08:35
IanStewart wrote:
jjewkes wrote:
I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.


I'd point out that while I've seen the occasional prep company question that asks about distances from points to lines, I've never, in ten years in this field, seen a real GMAT question that does. So while memorizing formulas like the one in the posts above might help you on a high school coordinate geometry test, it is extremely unlikely to make any difference on the GMAT.

And if you were given some point (a, b) and some line y = mx + c, and asked to find the perpendicular distance from (a,b) to the line, you could always:

* find the equation of the line perpendicular to y = mx + c which contains (a,b). That line has a slope of -1/m, and you can find its y-intercept by plugging in the point

* now find where the first line and its perpendicular intersect by solving the two equations together - that will be at some point (d,e)

* find the distance from (d,e) to (a,b) to get the answer

Following those steps, you can derive the formula in the posts above. You may well need to do one of the three things above in a question, but you're very unlikely to need to do all three, so I think it's vastly preferable to learn each concept individually rather than memorizing formulas that you'll most likely never have occasion to use.



Ian, I had this question on my Real GMAT last month. Of course, there could have some back-solving way, but I have not seen such and numbers were ugly.
Re: Distance to Line/Point on Coordinate Plane- please help!   [#permalink] 15 Oct 2010, 08:35
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