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Math: Coordinate Geometry

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Re: Math: Coordinate Geometry [#permalink] New post 21 Feb 2010, 03:00
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Bunuel wrote:
Slope of a Line

The slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line.


Image

Given two points (x_1,y_1) and (x_2,y_2) on a line, the slope m of the line is:

m=\frac{y_2-y_1}{x_2-x_1}

If the equation of the line is given in the Point-intercept form: y=mx+b, then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.

If the equation of the line is given in the General form:ax+by+c=0, then the slope is -\frac{a}{b} and the y intercept is -\frac{c}{b}.

SLOPE DIRECTION
The slope of a line can be positive, negative, zero or undefined.

Image

Positive slope
Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number. The line below has a slope of about +0.3, it goes up about 0.3 for every step of 1 along the x-axis.

Negative slope
Here, y decreases as x increases, so the line slopes downwards to the right. The slope will be a negative number. The line below has a slope of about -0.3, it goes down about 0.3 for every step of 1 along the x-axis.




Zero slope
Here, y does not change as x increases, so the line in exactly horizontal. The slope of any horizontal line is always zero. The line below goes neither up nor down as x increases, so its slope is zero.
Undefined slope
When the line is exactly vertical, it does not have a defined slope. The two x coordinates are the same, so the difference is zero. The slope calculation is then something like slope=\frac{15}{0} When you divide anything by zero the result has no meaning. The line above is exactly vertical, so it has no defined slope.

SLOPE AND QUADRANTS:

1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.

3. Every line (but the one crosses origin OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin (0,0) OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of 0, is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.

5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is x=a, where a is x-intercept.

6. For a line that crosses two points (x_1,y_1) and (x_2,y_2), slope m=\frac{y_2-y_1}{x_2-x_1}

7. If the slope is 1 the angle formed by the line is 45 degrees.

8. Given a point and slope, equation of a line can be found. The equation of a straight line that passes through a point (x_1, y_1) with a slope m is: y - y_1 = m(x - x_1)


A general question on Slope....

I know the an absolute value of a slope gives us how steep the line would be. And the sign gives us whether it is a rise or a fall...

But if we have a question like:
Line A has a slope -5 and Line B has a slope 4.... Which one of them has a greater slope? How do we handle this? Does this mean we consider the absolute values and then decide or answer.. (that is Line A)... or should we consider the signs too.. (i.e. Line B)...

Please advise!
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Re: Math: Coordinate Geometry [#permalink] New post 21 Feb 2010, 07:55
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jeeteshsingh wrote:
A general question on Slope....

I know the an absolute value of a slope gives us how steep the line would be. And the sign gives us whether it is a rise or a fall...

But if we have a question like:
Line A has a slope -5 and Line B has a slope 4.... Which one of them has a greater slope? How do we handle this? Does this mean we consider the absolute values and then decide or answer.. (that is Line A)... or should we consider the signs too.. (i.e. Line B)...

Please advise!


If the question is which one has the greater slope, then the answer would be: Line B, as 4>-5. As you correctly noted line A will be steeper than B, but the slope of B is positive and that of A is negative. We are comparing m1 with m2 not |m1| with |m2|.
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Re: Math: Coordinate Geometry [#permalink] New post 09 May 2010, 10:30
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There's one formula I'd like to add to the list:

Area of a triangle given the three vertices of the triangle (x1, y1), (x2, y2) and (x3, y3)

A = \frac{1}{2} * [x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]

To remember the order of variables just remember 1,2,3 then 2,3,1 and 3,1,2.

This would be useful in coordinate geometry in case the triangle does not rest on one of the axes and you know the three vertices.
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Re: Math: Coordinate Geometry [#permalink] New post 27 Jun 2010, 11:09
Great post, cleared the basics of the co-ordinate geometry.
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Re: Math: Coordinate Geometry [#permalink] New post 05 Jul 2010, 06:47
Thank you so much for this chapter! It's very well written. :)
One suggestion though - when we're actually doing GMAT question, and there is a fraction involved in the calculation, it is more often than not better to avoid converting it into decimal form until the absolute end of the question. Two reasons:
1. In Example #1 under Parellel lines section, it is not necessary to convert the slopes 14/29 and 20/-9 into decimal form. This is because the question requires us to figure out if the slopes are equal or not, and from the fraction form itself we can figure that out.
2. Often, you will be able to cancel out some parts of your fraction in a calculation that is to take place in the next step. For example, 9/2 is x. Find 2x. Answer: 9. (too easy example, but i hope u get the point.)
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Re: Math: Coordinate Geometry [#permalink] New post 17 Jul 2010, 14:25
Thanks So Much Man. A one point place for reviewing Coordinate Geometry. I have read most of these during EAMCET(Entrance Exam In A.P, India) time ;), now recollecting all, thanks to you. It would definitely take a long time to Google and learn all these, since most of the books dont cover so deep of a subject. Thanks Again and Keep on doing the great work.
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Re: Math: Coordinate Geometry [#permalink] New post 18 Jul 2010, 04:41
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For parabola if a = +ve it will open upwards
Open upwards means U....which is smiling
Attachment:
happy.jpg
happy.jpg [ 2.43 KiB | Viewed 22864 times ]



If a = -ve...it will open downwards
Open downwards means its unhappy
Attachment:
sad.jpg
sad.jpg [ 1.65 KiB | Viewed 22871 times ]

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Re: Math: Coordinate Geometry [#permalink] New post 02 Aug 2010, 08:17
Does anyone know how often the GMAT asks for anything beyond basic distance/slope? I have a hard time with these, especially rembering all the formulas, I have never seen a parabola question for example. Is it likely I would need to have this formula memorized?
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Re: Math: Coordinate Geometry [#permalink] New post 31 Aug 2010, 21:53
I guess this is typo mistake
Correct me if I am wrong

Your post is amazing
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Re: Math: Coordinate Geometry [#permalink] New post 30 Sep 2010, 05:40
Bunnel:

Suppose we have two lines ax+by+c = 0

and 2ax+2by+2c = 0 and question is- are they parallel? I know both are same lines, but do we can them parallel?

In DS question the answer of- are they parallel should be NO? or Yes.
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Re: Math: Coordinate Geometry [#permalink] New post 30 Sep 2010, 06:17
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Bunnel:

Suppose we have two lines ax+by+c = 0

and 2ax+2by+2c = 0 and question is- are they parallel? I know both are same lines, but do we can them parallel?

In DS question the answer of- are they parallel should be NO? or Yes.


Basically you are asking whether the line is parallel to itself. It depends how we define the word "parallel". I don't think that there is a consensus about this issue nor that this concept is tested on GMAT. So don't worry about it.
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Re: Math: Coordinate Geometry [#permalink] New post 30 Oct 2010, 08:56
Hi all,

Could you please explain how we can solve the below question.
If two lines are intersecting at point (10,27) and equation of one line is y=3x-3
What is the equation of another line.
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Re: Math: Coordinate Geometry [#permalink] New post 31 Oct 2010, 11:55
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to solve this que, u must need the slope of the another line... for example if these 2 lines are perpendicular or at an angle, say 30 degree.. then only u'll be able to find the equation of the second line.

so, if it a DS question, your ans sud be E ;) both the statement together r not sufficient

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diptich12 wrote:
Hi all,

Could you please explain how we can solve the below question.
If two lines are intersecting at point (10,27) and equation of one line is y=3x-3
What is the equation of another line.
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Re: Math: Coordinate Geometry [#permalink] New post 02 Nov 2010, 12:46
Just what i needed, thanks!
+1!
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Re: Math: Coordinate Geometry [#permalink] New post 03 Nov 2010, 05:05
really helpful post
thanks
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Re: Math: Coordinate Geometry [#permalink] New post 06 Nov 2010, 18:55
Seriously awesome. I think your coordinate geometry breakdown is better than MGMAT and Jeff Sackman's Total GMAT Math. This post helped me a ton. Thank you.
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Re: Math: Coordinate Geometry [#permalink] New post 11 Nov 2010, 14:45
Quote:
1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.

3. Every line (but the one crosses origin OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of , is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.

5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is , where a is x-intercept.



Hi Bunuel, Small corrections here- some of the words need to be intercepts and not intersect.

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Re: Math: Coordinate Geometry [#permalink] New post 12 Nov 2010, 07:58
awesome stuff!!
Can't possibly thank you enough
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Re: Math: Coordinate Geometry [#permalink] New post 31 Jan 2011, 08:56
hmmm. this is good work . very helpful .
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Re: Math: Coordinate Geometry [#permalink] New post 01 Apr 2011, 09:04
I had a question regarding this:
Q: Find the equation of a line whose x intercept is 5 and y intercept is 2.
Solution: Substituting the values in equation \frac{x}{a}+\frac{y}{b}=1 we'll get \frac{x}{5}+\frac{y}{2}=1 --> 5y+2x-10=0 OR if we want to write the equation in the slope-intercept form: y=-\frac{2}{5}x+2

Is that right? When you say x intercept is 5 then the points are (0,5) right? Same for y intercept is 2..(2,0).
So when you look at the equation in slope intercept form it should be
y = (-5/2)x + 5
Re: Math: Coordinate Geometry   [#permalink] 01 Apr 2011, 09:04
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