Matematické Fórum

Skrytý text:

just an intuitive approach, but i believe it works




We can try to solve (1) by looking for the intersections of the line and the circle with center   and
radius in plane , doing so for all .

Since the radius of the circle grows slower than the line is moving, solutions of (1) satisfy the condition
,   where    .

I think there is no computation needed to believe that the existence of numbers is undeniable in this case.
We can find them in this exercise by considering , then we find real solutions of (1):
.

Thus, if satisfies the equation (1), then .

If       or      , the line      and the circumference of relevant circle intersect each other in one point.
(it's not difficult to guess in which one it'll happen)

If      , the line intersects the circumference of relevant circle in two points, let's mark them  for instance,

and they satisfy  .

Now, let's try to find some helpful information by solving (2).


.

So, if      satisfies (2), then      and   .

The point is, that     itself has to satisfy the last two inequalities as well.

This result together with (3) implies following:

If      and      satisfy (1) and (2), then   .

Thus,
if      and      satisfy (1) and (2), then   .