Area of a triangle

Construct a unit circle $$ x^2+y^2 = 1 $$ between (0,1) and (1,0).
Then construct the hyp. $$ x^2-y^2 = 1$$ in the same interval.
You should have the ++ quadrant of a unit circle. and the upper half of an hyperbola, coincident at (0,1). Construct any line from (0,0) to the curve $$ x^2-y^2 = 1$$

Write the formula for the area of the triangle bounded by y=0; y=tan(theta) where theta is <pi/4, and subtended by the line from (0,0) to the hyperbola; and the hyperbolic curve, in terms of x and y.

Can you construct a line from (0.5,0) to the same point on the hyperbola as y=tan(theta) which is the upper right vertex of the triangle? How close can theta be to pi/2, and what limits this?

P,S, you may notice that you can't fit $$ x^2-y^2 = 1$$ into (0,1), (1,0); the hyp. lives 'outside' x=1; you need (0,1), (1,y).

A note about a line 'from' a point, if the point is (0,0) a line is all the points beyond it, so that the line has a single direction towards another point, which the line excludes - a line is a list with 3 elements.
The exclusion of the start and endpoints of any line is because the start is a reference that has to be independent so it has a successor, which is the first point in the line; the endpoint is the successor of the last point in the line.

(0,0) has to have a successor in any direction. a list for any line starts with p0, then p0' means p0' = p0 + p0'; p0'' = p0' + p0'', etc. the last point pN in the line has a successor pN'; the list is:

{p0,{p0',p0'',... ,pN},pN'} = {p0,{p1 p2 ... pN},pN'}.

If you colour the area outside the circle's ++ quadrant, up to x = 1, and y =1, it has an area too. So does the area outside the hyperbola, but it extends to infinity, so you need an infinite colour.

If you flip the quadrant left-to-right along the line x = 1, which is tangent to both curves, and draw another circle with origin at the focus of the hyperbola, can you find the ratio between it and the unit circle?

Note:

p0' means p0' = p0 + p0'

Click to expand...

doesn't imply addition of values, it's a successor function, it generates a position or a place. It says "the successive position of p0 is p0' = (p1)" not "add p0 to p0' and write the result in p1".
It generates the position or the next ordinal (zero for the origin, which is the head of the list), not a cardinal value.

It's not a triangle if one of the sides is the curve of a hyperbola.

Vkothii---

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