The law of inverse proportionality
expresses the law of inverse proportionality between
variables x and y. Variable y decreases by as many times as x increases. For m
> 0, x and y have the same sign, i.e. the graph is located in the first and
third quadrants of the coordinate axes; similarly, the graph is in the second
and fourth quadrants, for m < 0. As x approaches zero, the absolute value of
the fraction m/x becomes very large. Conversely, for large absolute values of
x} the ratio mix becomes small in absolute value.
Plotting the inverse proportionality
Plotting this curve
directly gives us Fig. 20, which shows curves (11) for various m, the full-line
curves corresponding to m > 0, and the broken curves to m < 0, the corresponding value of m being noted against each curve. We see that each of
the curves drawn, called rectangular hyperbolas, has infinite branches, the
points on a given branch approaching the coordinate axis X or Y with
indefinite increase of the abscissa x or the ordinate y, respectively.
These lines are called asymptotes to the hyperbola.
The coefficient m in equation (11) is completely defined if
any point M0 (X0,Y0) of the curve in question is given, since now:
equation (11) can now be written:
We draw new lines parallel to the axes through each pair of points of intersection lying on a given line of the pencil; the points of intersection of these new lines are then points of the rectangular hyperbola (Fig.21). This follows from the similarity of triangles 0RQ1 and 0SP1:Hence follows a graphical method for obtaining any required number of points of a rectangular hyperbola, given its asymptotes and any one point of it M0 (X0,Y0). Taking the asymptotes as coordinate axes, we draw an arbitrary pencil of lines
i.e. the point M0 (X0,Y0) lies on the
curve (12).
the source:
A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.
By: Fady tarek
0 comments:
Post a Comment