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Question Number 31569 by momo last updated on 10/Mar/18

(a, \frac{1}{a}), (b, \frac{1}{b}), (c, \frac{1}{c}), (d, \frac{1}{d}) a r e f o u r

d i s t i n c t p o i n t s o n a c i r c l e o f r a d i u s

i s 4 u n i t s t h e n a b c d i s e q u a l t o ?

Answered by MJS last updated on 10/Mar/18

center = (0; 0) :

x^{2} + y^{2} - 16 = 0

y = \frac{1}{x}

\Rightarrow x^{2} + \frac{1}{x^{2}} - 16 = 0

x^{4} - 16 x^{2} + 1 = 0

x = \sqrt{t}

t^{2} - 16 t + 1 = 0

t = 8 \pm \sqrt{64 - 1} = 8 \pm 3 \sqrt{7}

x_{1} = - \sqrt{8 + 3 \sqrt{7}}

x_{2} = - \sqrt{8 - 3 \sqrt{7}}

x_{3} = \sqrt{8 - 3 \sqrt{7}}

x_{4} = \sqrt{8 + 3 \sqrt{7}}

a b c d = x_{1} x_{2} x_{3} x_{4} = 1

center = (m; n)

{(x - m)}^{2} + {(y - n)}^{2} = r^{2}

y = \frac{1}{x}

{(x - m)}^{2} + {(\frac{1}{x} - n)}^{2} - r^{2} = 0

x^{2} - 2 m x + m^{2} + \frac{1}{x^{2}} - \frac{2 n}{x} + n^{2} - r^{2} = 0

x^{4} - 2 {m x}^{3} + (m^{2} + n^{2} - r^{2}) x^{2} - 2 n x + 1 = 0

\Rightarrow x_{1} x_{2} x_{3} x_{4} = 1

for x_{i} \in C

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