Find the lines through these points ?

Question

(1) Find the hyperbolic line through (1,2) and (3,4)

(2) Find the spherical line (great circle) through (1/2, 1/2, (1/2)^1/2) and (1,0,0)

I have been trying to solve this for 3 days with no luck. Please explain the steps. The textbook does not show an example and neither has a solution. This assignment is about to be due on Friday and I am panicking because there's no reference or clue how to do this

Answer 1

(1)

There exist infinitely many hyperbolas passing through points (1,2) and (3,4). I suppose that your task is to find any one of them.

If it is so, then we have a freedom to choose some of the parameters of the hyperbola.

I choose to find a hyperbola with one asymptote identical with x-axis, and the second asymptote parallel to y-axis.

Such hyperbola have an equation

y = m/(x – n)

where m and n are parameters to be found.

Now insert the points (1,2) and (3,4) to the equation of hyperbola:

2 = m/(1 – n)

4 = m/(3 – n)

Solution of this system of equation is

m = - 8

n = 5

so the equation of the hyperbola is

y = -8/(x – 5)

or in more elegant form  

y = 8/(5 - x)

See in the graph http://tinyurl.com/62loo9 that the hyperbola really passes through points (1,2) and (3,4).

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(2)

Note that the distance of both points (1/2, 1/2, (1/2)^1/2) and (1,0,0) from the point (0,0,0) is equal to 1.

Therefore our circle is the intersection of the sphere

x² + y² + z² = 0

and the plane

√(2) y - z = 0

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