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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

	$s o l v e x^{2} = r y a n d {(x - r)}^{2} + {(y - r)}^{2} = r^{2}$ ${(x - r)}^{2} + {(\frac{x^{2}}{r} - r)}^{2} = r^{2}$ $x^{2} - 2 x r + r^{2} + \frac{x^{4}}{r^{2}} - 2 x^{2} + r^{2} = r^{2}$ $\frac{x^{4}}{r^{2}} - x^{2} - 2 x r + r^{2} = 0$ $x^{4} - r^{2} x^{2} - 2 {x r}^{3} + r^{4} = 0$ $o n s o l v i n g w e g e t f o u r v a l u e o f x$ $s o c o r r e s p o n d i n g f o u r v a l u e o f y$ $t h u s w e c a n g e t c o i r i d i n a t e o f p o i n t A a n d C$ $o k l e t t r y t o s o l v e . . .$ $l e t \frac{x}{r} = k$ $\frac{x^{4}}{r^{4}} - \frac{x^{2}}{r^{2}} - \frac{2 x}{r} + 1 = 0$ $k^{4} - k^{2} - 2 k + 1 = 0$ $f (k) = k^{4} - k^{2} - 2 k + 1 = k^{4} - k^{2} - 2 k - 1 + 2 = k^{4} + 2 - {(k + 1)}^{2}$ $f (0) > 0$ $f (1) < 0$ $f (0.5) = \frac{1}{16} - \frac{1}{4} - \frac{2}{2} + 1 < 0$ $f (0.4) = 0.0256 + 2 - 1.96 > 0$ $s o v a l u e o f [0.5 > k > 0.4]$ $t h u s 0.5 > \frac{x}{r} > 0.4$ $s o 0.5 r > x > 0.4 r$ $t h u s x = (0.4 + ϵ) r ϵ = i s a s m a l l n u m b e r . . .$
	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

	$i c a n n o t s o l v e w i t h o u t n u m i r i c a l v a l u e . .$ $i f n u m i r i c a l v a l u e p r e s e n t i c a n u s e c a l c u l u s . .$ $b u t g e n e r a l e q n . . . i c a n n o t s o l v e . . .$
	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
	$(x−r)^2 +(y−r)^2 =r^2 (circle eq.) y=x^2 /r (parabola eq.) for x= x_A , x_C x^2 −2rx+(x^4 /r^2 )−2x^2 +r^2 =0 (x^4 /r^2 )−x^2 −2rx+r^2 =0 if x/r =t ⇒ t^4 −t^2 −2t+1=0 ....(i) let us assume equivalently (t^2 −pt+q)(t^2 +pt+(1/q))=0 ⇒ t^4 +(q+(1/q)−p^2 )t^2 +(pq−(p/q))t+1=0 comparing with (i) q+(1/q)= p^2 −1 and q−(1/q)=−(2/p) ⇒ (p^2 −1)^2 −(4/p^2 ) = 4 let us call p^2 =s ⇒ s(s−1)^2 −4s−4=0 let s−1= u ⇒ (u+1)u^2 −4(u+2)=0 ⇒ u^3 +u^2 −4u−8=0 let u=z−1/3 ⇒ z^3 −z^2 _− +(z/3)−(1/(27))+z^2 _− −((2z)/3)+(1/9) −4z+(4/3)−8=0 or z^3 −((13)/3)z−((178)/(27))=0 z= (((89)/(27))+(√(((89^2 )/(27^2 ))−((13^3 )/(27^2 )))) )^(1/3) +(((89)/(27))−(√((((89)/(27^2 ))−((13^3 )/(27^2 )))) )^(1/3) =(1/3){(89+6(√(159)))^(1/3) +(89−6(√(159)))^(1/3) now p^2 =s=u+1 = z+(2/3) since t_A >0 and t_C >0 let′s first choose (t^2 −pt+q)=0 with p_1 >0 p_1 =(√((2+(89+6(√(159)))^(1/3) +(89−6(√(159)))^(1/3) )/3)) ≈ 1.81226 (D/4)= (p^2 /4)−q as q= (((p^2 −1))/2)−(1/p) (D/4)=(1/2)+(1/p)−(p^2 /4) ≈ 0.23073 if it is positive then __________________________ with p_1 =(√((2+(89+6(√(159)))^(1/3) +(89−6(√(159)))^(1/3) )/3)) x_A =r( (p_1 /2)−(√((1/2)+(1/p_1 )−(p_1 ^2 /4)))) ⇒ x_A ≈ 0.4258 r , y_A =x_A ^2 /r x_C =r( (p_1 /2)+(√((1/2)+(1/p_1 )−(p_1 ^2 /4)))) ⇒ x_C ≈ 1.3865 r , y_C =x_C ^2 /r __________________________■$
	${(x - r)}^{2} + {(y - r)}^{2} = r^{2} (c i r c l e e q .)$ $y = x^{2} / r (p a r a b o l a e q .)$ $f o r x = x_{A}, x_{C}$ $x^{2} - 2 r x + \frac{x^{4}}{r^{2}} - 2 x^{2} + r^{2} = 0$ $\frac{x^{4}}{r^{2}} - x^{2} - 2 r x + r^{2} = 0$ $i f x / r = t$ $\Rightarrow t^{4} - t^{2} - 2 t + 1 = 0 . . . . (i)$ $l e t u s a s s u m e e q u i v a l e n t l y$ $(t^{2} - p t + q) (t^{2} + p t + \frac{1}{q}) = 0$ $\Rightarrow t^{4} + (q + \frac{1}{q} - p^{2}) t^{2} + (p q - \frac{p}{q}) t + 1 = 0$ $c o m p a r i n g w i t h (i)$ $q + \frac{1}{q} = p^{2} - 1 a n d q - \frac{1}{q} = - \frac{2}{p}$ $\Rightarrow {(p^{2} - 1)}^{2} - \frac{4}{p^{2}} = 4$ $l e t u s c a l l p^{2} = s$ $\Rightarrow s {(s - 1)}^{2} - 4 s - 4 = 0$ $l e t s - 1 = u$ $\Rightarrow (u + 1) u^{2} - 4 (u + 2) = 0$ $\Rightarrow u^{3} + u^{2} - 4 u - 8 = 0$ $l e t u = z - 1 / 3$ $\Rightarrow z^{3} - {\underline{z}}^{2} + \frac{z}{3} - \frac{1}{27} + {\underline{z}}^{2} - \frac{2 z}{3} + \frac{1}{9}$ $- 4 z + \frac{4}{3} - 8 = 0$ $o r z^{3} - \frac{13}{3} z - \frac{178}{27} = 0$ $z = {(\frac{89}{27} + \sqrt{\frac{89^{2}}{27^{2}} - \frac{13^{3}}{27^{2}}})}^{1 / 3}$ $+ {(\frac{89}{27} - \sqrt{(\frac{89}{27^{2}} - \frac{13^{3}}{27^{2}}})}^{1 / 3}$ $= \frac{1}{3} {{(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}$ $n o w p^{2} = s = u + 1 = z + \frac{2}{3}$ $s i n c e t_{A} > 0 a n d t_{C} > 0$ ${l e t}^{'} s f i r s t c h o o s e (t^{2} - p t + q) = 0$ $w i t h p_{1} > 0$ $p_{1} = \sqrt{\frac{2 + {(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}}{3}}$ $\approx 1.81226$ $\frac{D}{4} = \frac{p^{2}}{4} - q$ $a s q = \frac{(p^{2} - 1)}{2} - \frac{1}{p}$ $\frac{D}{4} = \frac{1}{2} + \frac{1}{p} - \frac{p^{2}}{4} \approx 0.23073$ $i f i t i s p o s i t i v e t h e n$ $__________________________$ $w i t h$ $p_{1} = \sqrt{\frac{2 + {(89 + 6 \sqrt{159})}^{1 / 3} + {(89 - 6 \sqrt{159})}^{1 / 3}}{3}}$ $x_{A} = r (\frac{p_{1}}{2} - \sqrt{\frac{1}{2} + \frac{1}{p_{1}} - \frac{p_{1}^{2}}{4}})$ $\Rightarrow x_{A} \approx 0.4258 r, y_{A} = x_{A}^{2} / r$ $x_{C} = r (\frac{p_{1}}{2} + \sqrt{\frac{1}{2} + \frac{1}{p_{1}} - \frac{p_{1}^{2}}{4}})$ $\Rightarrow x_{C} \approx 1.3865 r, y_{C} = x_{C}^{2} / r$ $__________________________◼$
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