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Question Number 135952 by Dwaipayan Shikari last updated on 17/Mar/21

Commented by Dwaipayan Shikari last updated on 17/Mar/21

$I t {i s A r c h i m e d e s}^{'} s N o t h i n g G r i n d e r . W e c a n m o v e t h i s b a c k$ $a n d f o r t h a n d s i d e w a y s . P r o v e t h a t t h e t i p o n v a r i o u s$ $p o s i t i o n s o n t h a t h a n d o f t h e w o d d e n g r i n d e r c r e a t e s$ $v a r i o u s s h a p e s o f e l l i p s e s$
Commented by Dwaipayan Shikari last updated on 17/Mar/21
Take a look at this animation https://en.wikipedia.org/wiki/File:Trammel_of_Archimedes_Large.gif
	Answered by mr W last updated on 17/Mar/21

	Commented by mr W last updated on 17/Mar/21

	$s a y$ $B (t, 0)$ $A (0, s)$ $s^{2} + t^{2} = p^{2}$ $x_{C} = \frac{q + p}{p} t \Rightarrow t = \frac{p}{p + q} x_{C}$ $y_{C} = - \frac{q}{p} s \Rightarrow s = - \frac{p}{q} y_{C}$ ${(\frac{p}{p + q} x_{C})}^{2} + {(- \frac{p}{q} y_{C})}^{2} = p^{2}$ $\frac{x_{C}^{2}}{{(p + q)}^{2}} + \frac{y_{C}^{2}}{q^{2}} = 1$ $o r$ $\frac{x^{2}}{{(p + q)}^{2}} + \frac{y^{2}}{q^{2}} = 1$ $i . e . t h e l o c u s o f C i s a n e l l i p s e w i t h$ $s e m i a x s e s :$ $a = p + q$ $b = q$
	Commented by Dwaipayan Shikari last updated on 17/Mar/21

	$T h a n k s s i r! i {d i d n}^{'} t t h i n k t h a t t h e p r o o f i s s i m p l e$
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