nimishasharma88 wrote:
m still confused....wat is the concept of slope?How do we know that AB has a negative slope?can sum1 plz expalin
Below is the snippet from GMAT Club Math book:
math-coordinate-geometry-87652.html --a must read for deep dive in coordinate geometry. This topic should explain the concept of slope:
Slope of a LineThe 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 DIRECTIONThe slope of a line can be positive, negative, zero or undefined.
Positive slopeHere, 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 slopeHere, 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 slopeHere, 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 slopeWhen 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: \(y - y_1 = m(x - x_1)\)
_________________
Thanks,
Prashant Ponde
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