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Question Number 80222 by mr W last updated on 01/Feb/20

Commented by mr W last updated on 01/Feb/20

$A r o u n d s t e e l p l a t e w i t h r a d i u s R h a s$ $a r o u n d h o l e w i t h r a d i u s r a s s h o w n .$ $W a s i s t h e m a x i m a l a r e a o f t h e e l l i p s e$ $w h i c h c a n b e c u t o f f f r o m t h e p l a t e ?$
Commented by behi83417@gmail.com last updated on 01/Feb/20

$dear master! please see my commenet on$ $You can't use 'macro parameter character #' in math mode$
	Answered by mr W last updated on 02/Feb/20

	Commented by mr W last updated on 02/Feb/20

	$p a r a m e t e r s o f e l l i p s e : a, b$ $c e n t e r o f e l l i p s e : C (0, c)$ $b - c = R - 2 r \Rightarrow c = b + 2 r - R$ $e q n . o f b i g c i r c l e :$ $x^{2} + y^{2} = R^{2}$ $e q n . o f e l l i p s e :$ $\frac{x^{2}}{a^{2}} + \frac{{(y - c)}^{2}}{b^{2}} = 1$ $i n t e r s e c t i o n f r o m c i r c l e a n d e l l i p s e :$ $\frac{R^{2} - y^{2}}{a^{2}} + \frac{{(y - c)}^{2}}{b^{2}} = 1$ $b^{2} (R^{2} - y^{2}) + a^{2} {(y - c)}^{2} = a^{2} b^{2}$ $(a^{2} - b^{2}) y^{2} - 2 {c a}^{2} y + a^{2} c^{2} - a^{2} b^{2} + R^{2} b^{2} = 0$ $d u e t o t a n g e n c y Δ = 0 :$ $c^{2} a^{4} - (a^{2} - b^{2}) (a^{2} c^{2} - a^{2} b^{2} + R^{2} b^{2}) = 0$ $a^{2} (a^{2} - b^{2} - R^{2} + c^{2}) + R^{2} b^{2} = 0$ $a^{4} - 4 r (R - r) a^{2} - 2 (R - 2 r) a^{2} b + R^{2} b^{2} = 0$ $\Rightarrow b = \frac{(R - 2 r) a^{2} + 2 a \sqrt{r (R - r) (R^{2} - a^{2})}}{R^{2}}$ $l e t P = a b$ $a^{6} - 4 r (R - r) a^{4} - 2 (R - 2 r) a^{3} P + R^{2} P^{2} = 0$ $6 a^{5} - 16 r (R - r) a^{3} - 2 (R - 2 r) (3 a^{2} P + a^{3} \frac{d P}{d a}) + 2 R^{2} P \frac{d P}{d a} = 0$ $f o r m a x i m u m e l l i p s e, \frac{d P}{d a} = 0$ $3 a^{2} - 8 r (R - r) = 3 (R - 2 r) b$ $3 a^{2} - 2 R^{2} = \frac{3 (R - 2 r)}{2 r (R - r)} a \sqrt{r (R - r) (R^{2} - a^{2})}$ $[9 + \frac{9 {(R - 2 r)}^{2}}{4 r (R - r)}] a^{4} - [12 + \frac{9 {(R - 2 r)}^{2}}{4 r (R - r)}] R^{2} a^{2} + 4 R^{4} = 0$ $w i t h μ = \frac{9 {(R - 2 r)}^{2}}{4 r (R - r)}$ $\Rightarrow (9 + μ) {(\frac{a}{R})}^{4} - (12 + μ) {(\frac{a}{R})}^{2} + 4 = 0$ $\Rightarrow \frac{a}{R} = \sqrt{\frac{12 + μ \pm \sqrt{μ (8 + μ)}}{2 (9 + μ)}}$ $(‘ ‘ +^{″} i f R > 2 r, ‘ ‘ -^{″} i f R < 2 r)$
	Commented by mr W last updated on 02/Feb/20

	Commented by mr W last updated on 02/Feb/20

	Commented by mr W last updated on 02/Feb/20

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