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Question Number 88310 by ajfour last updated on 09/Apr/20

Commented by ajfour last updated on 09/Apr/20

$Q . 88272 (r e v i s i t e d)$
	Answered by ajfour last updated on 10/Apr/20
	$h^2 +k^2 =r^2 y^2 =b^2 (1−(((x−a)^2 )/a^2 )) [ellipse] x^2 +y^2 −2hx−2ky=0 [circle] (x^2 +y^2 −2hx)^2 =4(r^2 −h^2 )y^2 ⇒ {x^2 +b^2 (1−(((x−a)^2 )/a^2 ))−2hx}^2 = 4(r^2 −h^2 )b^2 (1−(((x−a)^2 )/a^2 )) roots of above eq. are x_1 =0 , x_2 =2h, x_3 =x_4 =t first we eliminate x=0, then determine h(r). then eliminate t and eventually find r. .... let (x/a)=X, (h/a)=s, (b^2 /a^2 )=q, (t/a)=m, (r/a)=R . {X^2 +q(2X−X^2 )−2sX}^2 = 4(R^2 −s^2 )q(2X−X^2 ) ⇒ X{X+q(2−X)−2s}^2 =4q(R^2 −s^2 )(2−X) roots are now X=2s, m,m. lets put X=2s ⇒qs(1−s)=(R^2 −s^2 ) ⇒ R^2 =(1−q)s^2 +qs ...(I) And R^2 −s^2 =qs(1−s) ...(II) now generally (1−q)^2 X^3 +4(1−q)(q−s)X^2 +4(q−s)^2 X +4q(R^2 −s^2 )X−8q(R^2 −s^2 )=0 ⇒ (1−q)^2 X^3 +4(1−q)(q−s)X^2 +4{(q−s)^2 +q(R^2 −s^2 )}X −8q(R^2 −s^2 )=0 2m+2s=−((4(q−s))/((1−q))) ...(i) 2m^2 s=((8q(R^2 −s^2 ))/((1−q)^2 )) ...(ii) ⇒ s{((2(s−q))/((1−q)))−s}^2 =((4q(R^2 −s^2 ))/((1−q)^2 )) now using (II) ⇒ s{((2(s−q))/((1−q)))−s}^2 =((4q(qs)(1−s))/((1−q)^2 )) ⇒ {((2(s−q))/((1−q)))−s}^2 =((4q^2 (1−s))/((1−q)^2 )) ⇒{(1+q)s−2q}^2 =4q^2 (1−s) (1+q)^2 s^2 −4qs=0 and as s≠0 s=((4q)/((1+q)^2 )) Now using (I) ⇒ R=(√((1−q)s^2 +qs )) ⇒ R = (√(((16(1−q)q^2 )/((1+q)^4 ))+((4q^2 )/((1+q)^2 )))) or R=((2q(√((1−q)^2 +4)))/((1+q)^2 )) for example if a=4, b=3 q=(9/(16)), R=((18(√(1073)))/(625)) ≈ 0.9434 ...$
	$h^{2} + k^{2} = r^{2}$ $y^{2} = b^{2} (1 - \frac{{(x - a)}^{2}}{a^{2}}) [e l l i p s e]$ $x^{2} + y^{2} - 2 h x - 2 k y = 0 [c i r c l e]$ ${(x^{2} + y^{2} - 2 h x)}^{2} = 4 (r^{2} - h^{2}) y^{2}$ $\Rightarrow$ ${x^{2} + b^{2} (1 - \frac{{(x - a)}^{2}}{a^{2}}) - 2 h x}^{2}$ $= 4 (r^{2} - h^{2}) b^{2} (1 - \frac{{(x - a)}^{2}}{a^{2}})$ $r o o t s o f a b o v e e q . a r e$ $x_{1} = 0, x_{2} = 2 h, x_{3} = x_{4} = t$ $f i r s t w e e l i m i n a t e x = 0,$ $t h e n d e t e r m i n e h (r) .$ $t h e n e l i m i n a t e t a n d$ $e v e n t u a l l y f i n d r .$ $. . . .$ $l e t \frac{x}{a} = X, \frac{h}{a} = s, \frac{b^{2}}{a^{2}} = q, \frac{t}{a} = m,$ $\frac{r}{a} = R .$ ${X^{2} + q (2 X - X^{2}) - 2 s X}^{2}$ $= 4 (R^{2} - s^{2}) q (2 X - X^{2})$ $\Rightarrow X {X + q (2 - X) - 2 s}^{2}$ $= 4 q (R^{2} - s^{2}) (2 - X)$ $r o o t s a r e n o w X = 2 s, m, m .$ $l e t s p u t X = 2 s$ $\Rightarrow q s (1 - s) = (R^{2} - s^{2})$ $\Rightarrow R^{2} = (1 - q) s^{2} + q s . . . (I)$ $A n d R^{2} - s^{2} = q s (1 - s) . . . (I I)$ $n o w g e n e r a l l y$ ${(1 - q)}^{2} X^{3} + 4 (1 - q) (q - s) X^{2} + 4 {(q - s)}^{2} X$ $+ 4 q (R^{2} - s^{2}) X - 8 q (R^{2} - s^{2}) = 0$ $\Rightarrow$ ${(1 - q)}^{2} X^{3} + 4 (1 - q) (q - s) X^{2}$ $+ 4 {{(q - s)}^{2} + q (R^{2} - s^{2})} X$ $- 8 q (R^{2} - s^{2}) = 0$ $2 m + 2 s = - \frac{4 (q - s)}{(1 - q)} . . . (i)$ $2 m^{2} s = \frac{8 q (R^{2} - s^{2})}{{(1 - q)}^{2}} . . . (i i)$ $\Rightarrow s {\frac{2 (s - q)}{(1 - q)} - s}^{2} = \frac{4 q (R^{2} - s^{2})}{{(1 - q)}^{2}}$ $n o w u s i n g (I I)$ $\Rightarrow s {\frac{2 (s - q)}{(1 - q)} - s}^{2} = \frac{4 q (q s) (1 - s)}{{(1 - q)}^{2}}$ $\Rightarrow {\frac{2 (s - q)}{(1 - q)} - s}^{2} = \frac{4 q^{2} (1 - s)}{{(1 - q)}^{2}}$ $\Rightarrow {(1 + q) s - 2 q}^{2}$ $= 4 q^{2} (1 - s)$ ${(1 + q)}^{2} s^{2} - 4 q s = 0$ $a n d a s s \neq 0$ $s = \frac{4 q}{{(1 + q)}^{2}}$ $N o w u s i n g (I)$ $\Rightarrow R = \sqrt{(1 - q) s^{2} + q s}$ $\Rightarrow R = \sqrt{\frac{16 (1 - q) q^{2}}{{(1 + q)}^{4}} + \frac{4 q^{2}}{{(1 + q)}^{2}}}$ $o r R = \frac{2 q \sqrt{{(1 - q)}^{2} + 4}}{{(1 + q)}^{2}}$ $f o r e x a m p l e i f a = 4, b = 3$ $q = \frac{9}{16}, R = \frac{18 \sqrt{1073}}{625} \approx 0.9434$ $. . .$
	Commented by mr W last updated on 10/Apr/20

	${i t}^{'} s p e r f e c t l y s o l v e d!$
	Commented by ajfour last updated on 10/Apr/20

	$g r e a t, t h a n k s f o r t h e c o n f i r m a t i o n,$ $S i r, I^{'} m t o o g l a d i c u d s o l v e i t .$
	Commented by mr W last updated on 09/Apr/20

	${t h a t}^{'} s r i g h t s i r .$
	Commented by ajfour last updated on 10/Apr/20

	$t h a n k s t o m j S S i r .$
	Commented by ajfour last updated on 10/Apr/20

	Commented by MJS last updated on 10/Apr/20

	$I tried the same path I used with the parabola$ $and also get R = \frac{18 \sqrt{1073}}{625} for \frac{a}{b} = \frac{4}{3} but there$ $should be a scaling factor, I calculated with$ $a = 1 \land b = \frac{3}{4}$
	Commented by ajfour last updated on 10/Apr/20

	$s o o u r a n s w e r s h o u l d b e r i g h t$ $t h e n . . ? r e q u i r e d r a d i u s r = a R, S i r .$
	Commented by mr W last updated on 10/Apr/20

	$i c h e c k e d m y r e s u l t a g a i n a n d f o u n d$ $a n e r r o r . n o w w e h a v e t h e s a m e r e s u l t .$
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