Question-54447

Question Number 54447 by ajfour last updated on 03/Feb/19

Commented by ajfour last updated on 14/Feb/19

F i n d c o o r d i n a t e s o f A a n d C .

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Feb/19

s o l v e x^{2} = r y a n d {(x - r)}^{2} + {(y - r)}^{2} = r^{2}

{(x - r)}^{2} + {(\frac{x^{2}}{r} - r)}^{2} = r^{2}

x^{2} - 2 x r + r^{2} + \frac{x^{4}}{r^{2}} - 2 x^{2} + r^{2} = r^{2}

\frac{x^{4}}{r^{2}} - x^{2} - 2 x r + r^{2} = 0

x^{4} - r^{2} x^{2} - 2 {x r}^{3} + r^{4} = 0

o n s o l v i n g w e g e t f o u r v a l u e o f x

s o c o r r e s p o n d i n g f o u r v a l u e o f y

t h u s w e c a n g e t c o i r i d i n a t e o f p o i n t A a n d C

o k l e t t r y t o s o l v e \dots

l e t \frac{x}{r} = k

\frac{x^{4}}{r^{4}} - \frac{x^{2}}{r^{2}} - \frac{2 x}{r} + 1 = 0

k^{4} - k^{2} - 2 k + 1 = 0

f (k) = k^{4} - k^{2} - 2 k + 1 = k^{4} - k^{2} - 2 k - 1 + 2 = k^{4} + 2 - {(k + 1)}^{2}

f (0) > 0

f (1) < 0

f (0.5) = \frac{1}{16} - \frac{1}{4} - \frac{2}{2} + 1 < 0

f (0.4) = 0.0256 + 2 - 1.96 > 0

s o v a l u e o f [0.5 > k > 0.4]

t h u s 0.5 > \frac{x}{r} > 0.4

s o 0.5 r > x > 0.4 r

t h u s x = (0.4 + ϵ) r ϵ = i s a s m a l l n u m b e r \dots

Commented by ajfour last updated on 03/Feb/19

w h y n o t s o l v e i t, S i r ?

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Feb/19

i c a n n o t s o l v e w i t h o u t n u m i r i c a l v a l u e . .

i f n u m i r i c a l v a l u e p r e s e n t i c a n u s e c a l c u l u s . .

b u t g e n e r a l e q n \dots i c a n n o t s o l v e \dots

Commented by ajfour last updated on 03/Feb/19

i t c a n b e c o n v e r t e d t o o n e l i k e

n u m e r i c a l v a l u e, b u t s t i l l l e t u s n o t

u s e c a l c u l a t o r .

Commented by ajfour last updated on 03/Feb/19

t h a n k s f o r t r y i n g, S i r .

Answered by ajfour last updated on 04/Feb/19

{(x - r)}^{2} + {(y - r)}^{2} = r^{2} (c i r c l e e q .)

y = x^{2} / r (p a r a b o l a e q .)

f o r x = x_{A}, x_{C}

x^{2} - 2 r x + \frac{x^{4}}{r^{2}} - 2 x^{2} + r^{2} = 0

\frac{x^{4}}{r^{2}} - x^{2} - 2 r x + r^{2} = 0

i f x / r = t

\Rightarrow t^{4} - t^{2} - 2 t + 1 = 0 \dots . (i)

l e t u s a s s u m e e q u i v a l e n t l y

(t^{2} - p t + q) (t^{2} + p t + \frac{1}{q}) = 0

\Rightarrow t^{4} + (q + \frac{1}{q} - p^{2}) t^{2} + (p q - \frac{p}{q}) t + 1 = 0

c o m p a r i n g w i t h (i)

q + \frac{1}{q} = p^{2} - 1 a n d q - \frac{1}{q} = - \frac{2}{p}

\Rightarrow {(p^{2} - 1)}^{2} - \frac{4}{p^{2}} = 4

l e t u s c a l l p^{2} = s

\Rightarrow s {(s - 1)}^{2} - 4 s - 4 = 0

l e t s - 1 = u

\Rightarrow (u + 1) u^{2} - 4 (u + 2) = 0

\Rightarrow u^{3} + u^{2} - 4 u - 8 = 0

l e t u = z - 1 / 3

\Rightarrow z^{3} - {\underline{z}}^{2} + \frac{z}{3} - \frac{1}{27} + {\underline{z}}^{2} - \frac{2 z}{3} + \frac{1}{9}

- 4 z + \frac{4}{3} - 8 = 0

o r z^{3} - \frac{13}{3} z - \frac{178}{27} = 0

z = {(\frac{89}{27} + \sqrt{\frac{89^{2}}{27^{2}} - \frac{13^{3}}{27^{2}}})}^{1 / 3}

+ {(\frac{89}{27} - \sqrt{(\frac{89}{27^{2}} - \frac{13^{3}}{27^{2}}})}^{1 / 3}

= \frac{1}{3} {{(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}

n o w p^{2} = s = u + 1 = z + \frac{2}{3}

s i n c e t_{A} > 0 a n d t_{C} > 0

{l e t}^{'} s f i r s t c h o o s e (t^{2} - p t + q) = 0

w i t h p_{1} > 0

p_{1} = \sqrt{\frac{2 + {(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}}{3}}

\approx 1.81226

\frac{D}{4} = \frac{p^{2}}{4} - q

a s q = \frac{(p^{2} - 1)}{2} - \frac{1}{p}

\frac{D}{4} = \frac{1}{2} + \frac{1}{p} - \frac{p^{2}}{4} \approx 0.23073

i f i t i s p o s i t i v e t h e n

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w i t h

p_{1} = \sqrt{\frac{2 + {(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}}{3}}

x_{A} = r (\frac{p_{1}}{2} - \sqrt{\frac{1}{2} + \frac{1}{p_{1}} - \frac{p_{1}^{2}}{4}})

\Rightarrow x_{A} \approx 0.4258 r, y_{A} = x_{A}^{2} / r

x_{C} = r (\frac{p_{1}}{2} + \sqrt{\frac{1}{2} + \frac{1}{p_{1}} - \frac{p_{1}^{2}}{4}})

\Rightarrow x_{C} \approx 1.3865 r, y_{C} = x_{C}^{2} / r

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