Conic Sections? Continued Fractions?

Algebra Level 3

The curve described by the continued fraction

x + x x + x x + x x + x x + x ⋱ \large x+\frac{x}{x+\frac{x}{x+\frac{x}{x+\frac{x}{x+\frac{x}{\ddots}}}}}

coincides with the curve described by a conic section C C . What type of conic section is C C ? What is the equation of C C ?

If the centre of C C is at a point ( x , y ) (x, y) , type x + y x+y as your answer.


The answer is -3.

Andrei Li by Andrei Li , 16, Canada

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1 solution

Solve for s s where

s = x + x s s = x + \dfrac{x}{s}

and get

s = 1 2 ( x − x 2 + 4 x ) s = \frac{1}{2}(x - \sqrt{x^2+4x})
s = 1 2 ( x + x 2 + 4 x ) s = \frac{1}{2}(x + \sqrt{x^2+4x})

which is a hyperbola, the asymptotes being

y = − 1 y = -1
y = x + 1 y = x + 1

which meet at ( − 2 , − 1 ) (-2, -1) , the center of this hyperbola

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