Question

Find the equations of the line in the half-plane model of the hyperbolic plane that passes through the points (a) (1,1) and (1,6). (b) (2, 2) and (4,4)

Answer 1

PRELIMINARIES:-

In a hyperbolic half-plane model , we use the following definitions ;

h-point: any point A(x,y) where y>0.

h-lines: (a) the vertical lines and

(b) the semi circles which have their center on the x-axis

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FINDING EQUATION OF LINE IN HYPERBOLIC HALF-PLANE MODEL:

We know that , a semi circle with center on the x-axis has equation of the form of

x²  + y²+ a.x = b-----------------(1)

Now , using equation we find the equations of the line in the hyperbolic half plane model

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Answer:(a)

Given points are (1,1) and(1,6).

Let A(1,1) and B(1,6) be the given points.

The line AB is a vertical ray , hence the equation of the h-line is x=1, y>0.

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Answer :(b)

Given points are (2,2) and (4,4).

Let these points be A(2,2) and B(4,4).

Since we are to find the equation of h-line passing through these two points , these two points must satisfy the equation    x²  + y²+ a.x = b , as given in (1).

Now putting the coordinates of A and B in the equation (1) we get,

2² + 2² + 2.a = b -------- (2)

and 4² + 4² + 4.a = b -------- (3)

solving equations (2) and (3) we get a=-12 and b=-16

putting these values of a and b in equation (1) we get the required equation of line in the half plane model as

x² + y² -12.x = -16

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