Searching the intersections of

and (a is real number)

by algebraic approach

by Kaori Tabeta


Substance y = ax for

Then

When x=0 the equations is conclusion with y=0. Therefore, if a is any number, there is the intersection of origin.

When x not = 0 then

If

therefore a=1

Then ,substance a=1 for

Then 0=3

It's not true. Therefore, as a=1, there is only intersection of origin.

If a not =1

 

And

The coordinates of the intersections without the origin are below.

The only sings differ from the two coordinates, so these two points are the central symmetry with the central point of origin.

For these coordination numbers are the real number:

By the way

Therefore

For

and so

or

and so

Summary

As , there is the only intersection of the origin.

As or ,there are three intersections.

The intersections are

The two intersections of the first are the central symmetry with the center point of (0,0).


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