Another way to parametrize the circle x2 + y2 = 1 is by the following method, ca

Question

Another way to parametrize the circle x2 + y2 = 1 is by the following method, called stere?graphic projection. Note that the po (0, 1) is on the circle. Find the equation of the line of slope t that passes through (0, 1). Find the po (x, y) on the circle where your line in part (a) ersects the circle. Your answer to part (b) gives you expressions for x and y in terms of t, so write down parametric equations of the circle. [This is a rational parametrization since x and y are given by rational functions of t.] From your answer in part (c), you'll see that only one po is not represented with this parametrization. What po is it?

Explanation / Answer

a) y = t*x+c, the line passes through 0,1 so putting in eq weg et

1 = t*0+c ==> c =1

so eq of line is y = tx+1

b)x^2+(tx+1)^2 = 1

==> x^2+t^2x^2+1+2tx = 1

==> x^2(1+t^2) +2tx = 0

==> x = 0, -2t/(1+t^2)

y = 1, -2t^2/(1+t^2)

point is (-2t/(1+t^2),  -2t^2/(1+t^2))

c) 4t^2/(1+t^2)^2 + 4t^4/(1+t^2)^2 = 1

==> 4t^2(1+t^2)/(1+t^2)^2 = 1

==> 4t^2/(1+t^2) = 1

==> 4t^2 = 1+t^2

==> 3t^2-1 = 0

d)

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