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Re: Math: Coordinate Geometry [#permalink]
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.
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.
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! _________________
Cheers! JT........... If u like my post..... payback in Kudos!!
|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice| |For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|
Re: Math: Coordinate Geometry [#permalink]
21 Feb 2010, 07:55
Expert's post
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|. _________________
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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If you feel like you're under control, you're just not going fast enough. A goal without a plan is just a wish. You can go higher, you can go deeper, there are no boundaries above or beneath you.
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.
Re: Math: Coordinate Geometry [#permalink]
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? _________________
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. _________________
Re: Math: Coordinate Geometry [#permalink]
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.
Re: Math: Coordinate Geometry [#permalink]
31 Oct 2010, 11:55
1
This post received KUDOS
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
If u like the post, please consider KUDOS.. Thanks
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.
Re: Math: Coordinate Geometry [#permalink]
06 Nov 2010, 18:55
1
This post received KUDOS
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.
Re: Math: Coordinate Geometry [#permalink]
11 Nov 2010, 14:45
1
This post received KUDOS
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.
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
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