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Re: Math: Coordinate Geometry [#permalink]
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Hi guys,
something I am missing.. why do we do a minus Xc (which is 12) as below for parallel lines???? is it always the case???

Slope AB=\frac{20-7}{5-30}=-052

For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by it's equation in point-slope form form:

y=-0.52(x-12)+10 --> y=-0.52x+16.24
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Re: Math: Coordinate Geometry [#permalink]
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Example # 2.
Q: Define a line passing through the point E and perpendicular to a line passing through the points C and D on the graph above.
Solution: The point E is on the y-axis and so is the y-intercept of the desired line. Once we know the slope of the line, we can express it using its equation in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept.

First find the slope of line CD:
Slope CD=\frac{24-4}{22-31}=\frac{20}{-9}=-2.22

The line we seek will have a slope which is the negative reciprocal of:
-\frac{1}{-2.22}=0.45

Since E is on the Y-axis, we know that the intercept is 10. Plugging these values into the line equation, the line we need is described by the equation

y = 0.45x + 10

What if E was not on the Y-axis???
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Re: Math: Coordinate Geometry [#permalink]
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defoue wrote:
Hi guys,
something I am missing.. why do we do a minus Xc (which is 12) as below for parallel lines???? is it always the case???

Slope AB=\frac{20-7}{5-30}=-052

For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by it's equation in point-slope form form:

y=-0.52(x-12)+10 --> y=-0.52x+16.24


The equation of a straight line that passes through a point \(P_1(x_1, y_1)\) with a
slope \(m\) is:

\(y-y_1=m(x-x_1)\)

We calculated the slope \(m=-0.52\), and have the point \(C(12,10)\). substituting the values in the equation above we get: \(y-10=-0.52(x-12)\) or \(y=-0.52(x-12)+10\) as written.

Hope it's clear.
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Re: Math: Coordinate Geometry [#permalink]
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defoue wrote:
Example # 2.
Q: Define a line passing through the point E and perpendicular to a line passing through the points C and D on the graph above.
Solution: The point E is on the y-axis and so is the y-intercept of the desired line. Once we know the slope of the line, we can express it using its equation in slope-intercept form y=mx+b, where m is the slope and b is the y-intercept.

First find the slope of line CD:
Slope CD=\frac{24-4}{22-31}=\frac{20}{-9}=-2.22

The line we seek will have a slope which is the negative reciprocal of:
-\frac{1}{-2.22}=0.45

Since E is on the Y-axis, we know that the intercept is 10. Plugging these values into the line equation, the line we need is described by the equation

y = 0.45x + 10

What if E was not on the Y-axis???


Slope of line segment CD would be calculated in the same way. Slope of perpendicular line would also be calculated in the same way.

So we have the point \(E(x_1,y_1)\) (no matter whether it's on y axis or not) and the slope \(m\) of a line which passes through this point.

And again: The equation of a straight line that passes through a point \(P_1(x_1, y_1)\) with a
slope \(m\) is:

\(y-y_1=m(x-x_1)\)

Only thing is left to substitute the values we have (point/slope) in above equation.

Hope it helps.
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Re: Math: Coordinate Geometry [#permalink]
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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!
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Re: Math: Coordinate Geometry [#permalink]
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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]
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]
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gurpreetsingh wrote:
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]
When I have two points in a coordinate system e.g. (2,3) and (6,7) that pass through a line how do I know which number is x1 and which is x2 when calculating slope. Also, the same for the y1 and y2?
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Re: Math: Coordinate Geometry [#permalink]
Wonderful! Excellent work!

A small remark at "7. If the slope is 1 the angle formed by the line is 45 degrees."
Angle formed by the line with who?
The slope of a line is a real number equal to the tangent of the angle formed by the line with the positive x axis.
For the GMAT, no need to know about tangent (trigonometry), just to understand which angle we are talking about.
If the angle is acute, the slope (tangent of the angle) is positive. If the angle is obtuse, the slope (tangent of the angle) is negative.
There is a one-to-one correspondence between angles and the real numbers representing their so-called tangent values.
So, a line with slope 1 forms an angle of 45 degrees and a line with slope -1 forms an angle of 135 degrees with the positive x axis.
If the angle is 60 degrees, the slope is \(\sqrt{3}\), and \(-\sqrt{3}\) if the angle is 120 degrees, etc.

A suggestion: wouldn't be better to define the slope somewhere before the equations? I am sure everybody knows what's the slope of a line, just for the coherence.
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Stiv wrote:
When I have two points in a coordinate system e.g. (2,3) and (6,7) that pass through a line how do I know which number is x1 and which is x2 when calculating slope. Also, the same for the y1 and y2?


It doesn't matter: \(slope=\frac{7-3}{6-2}=\frac{3-7}{2-6}=1\).
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Re: Math: Coordinate Geometry [#permalink]
HI
I have a basic doubt , which deals with concept
Previously , I have learnt that a line that has rising slope when moving from - ve X to + ve X will have +ve slope.
But I found in the original post of BUNNUEL that a line with - ve slope will always be in 2nd or 4th quadrant. I feel that these two statements are contradictory. We can have a line with + ve slope even if it is in 2nd quadrant. Than how can we come to conclusion that whenever we encounter a slope with - ve sign , thn it must either lie in 2nd or lie in 4 th quadrant. As we can have lines in 1st quadrant with - ve slope also........
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Re: Math: Coordinate Geometry [#permalink]
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Archit143 wrote:
HI
I have a basic doubt , which deals with concept
Previously , I have learnt that a line that has rising slope when moving from - ve X to + ve X will have +ve slope.
But I found in the original post of BUNNUEL that a line with - ve slope will always be in 2nd or 4th quadrant. I feel that these two statements are contradictory. We can have a line with + ve slope even if it is in 2nd quadrant. Than how can we come to conclusion that whenever we encounter a slope with - ve sign , thn it must either lie in 2nd or lie in 4 th quadrant. As we can have lines in 1st quadrant with - ve slope also........


Responding to a pm:

A line with +slope MUST lie in the 1st and 3rd quadrant. It can also lie in either the 2nd or the 4th quadrant or it may not lie in both 2nd and 4th (if it passes through the center). But it must lie in 1st as well as the 3rd quadrant.

A line with -ve slope MUST lie in the 2nd and 4th quadrant. It can also lie in either 1st or 3rd quadrant or it may not lie in both 1st and 3rd (if it passes through the center). But it must lie in 2nd as well as the 4th quadrant.

Draw some lines with +ve/-ve slopes to figure this out.
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Re: Math: Coordinate Geometry [#permalink]
So for a question where it is given that slope is -1/6, Than how can be sure that line intersects 2nd quad, I found this question on GMAT prep......
in-the-rectangular-coordinate-system-shown-above-does-the-90635.html

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Archit143 wrote:
So for a question where it is given that slope is -1/6, Than how can be sure that line intersects 2nd quad, I found this question on GMAT prep......
in-the-rectangular-coordinate-system-shown-above-does-the-90635.html

Archit


As discussed, line with negative slope (-1/6<0) MUST intersect quadrants II and IV.
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Re: Math: Coordinate Geometry [#permalink]
HI Bunuel,
As karishma has also mentioned that a line with -ve slope can also be present in 1st quadrant than how can we be so sure that it is intersecting in 2nd quadrant,

Moreover as per the theory it must intersect in 2nd and 4th quadrant ...than as per statement 1 there are two possibilities.....line intersecting in in quad 2 and 4.....But we must have only one answer form the statement, for it to be correct answer..................

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Archit143 wrote:
HI Bunuel,
As karishma has also mentioned that a line with -ve slope can also be present in 1st quadrant than how can we be so sure that it is intersecting in 2nd quadrant,

Moreover as per the theory it must intersect in 2nd and 4th quadrant ...than as per statement 1 there are two possibilities.....line intersecting in in quad 2 and 4.....But we must have only one answer form the statement, for it to be correct answer..................

Archit


No, that's not what she said.

If the slope of a line is negative, line WILL intersect quadrants II and IV in ANY case. If X and Y intersects are positives, line ALSO intersects the quadrant I, if negative line ALSO intersects the quadrant quadrant III.
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