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Question Number 186835 by ajfour last updated on 11/Feb/23

Commented by ajfour last updated on 11/Feb/23

$C u r v e i s y = x^{3} - x$ $H o r i z o n t a l l i n e s y = 0 a n d y = c$ $S a y a r o o t o f c u r v e b e x = p .$ $A (p, 0)$ $F r o m A a t a n g e n t i s d r a w n t o$ $c u r v e; t o u c h e s a t Q (q, q^{3} - q)$ $s a y e q u a t i o n b e$ $y = (3 q^{2} - 1) (x - p)$ $q^{3} - q = (3 q^{2} - 1) (q - p)$ $L e t a n o r m a l t o c u r v e f r o m Q$ $a n d a t Q i n t e r s e c t a t S (s, s^{3} - s)$ $L e t s \neq q$ $E q u a t i o n o f s u c h a n o r m a l i s$ $y - (q^{3} - q) = - \frac{(x - q)}{(3 q^{2} - 1)}$ $\Rightarrow s^{3} - s - q^{3} + q = - \frac{(s - q)}{3 q^{2} - 1}$ $\Rightarrow s^{2} + q^{2} + q s - 1 = \frac{1}{(1 - 3 q^{2})}$ $q^{3} - q = (3 q^{2} - 1) (q - p)$ $p^{3} = p + c$ $o r$ $s^{3} - s - (3 q^{2} - 1) (q - p) = \frac{s - q}{3 q^{2} - 1}$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ $w h i l e i f s = q$ $a n d w e f i n d w h e r e s u c h a n o r m a l$ $i n t e r s e c t s t h e c u r v e a g a i n, t h e n$ $(x^{3} - x) - (q^{3} - q) = - \frac{(x - q)}{(3 q^{2} - 1)}$ $\Rightarrow (x^{2} + q^{2} + q x - 1) (3 q^{2} - 1) + 1 = 0$ $&$ $q^{3} - q = (3 q^{2} - 1) (q - p)$ $w h i l e p^{3} = p + c$ $. . . .$
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