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Question Number 186803 by ajfour last updated on 10/Feb/23

	Answered by ajfour last updated on 10/Feb/23

	$C u r v e i s y = x^{3} - x$ $p^{3} - p = c$ $m = 3 s^{2} - 1 = \frac{p - h}{k}$ $T a n g e n t :$ $y - k = (3 s^{2} - 1) (x - h)$ $E q a t i o n f o r r o o t s o f I n t e r s e c t i o n$ $o f t a n g e n t l i n e w i t h c u r v e :$ $x^{3} - 3 s^{2} x - k + (3 s^{2} - 1) h = 0$ $A s x = s$ $2 s^{3} + k = (3 s^{2} - 1) h$ $2 s^{3} - 3 {h s}^{2} + h + k = 0$ $l e t s = t + q$ $2 (t^{3} + 3 {q t}^{2} + 3 q^{2} t + q^{3})$ $- 3 h (t^{2} + 2 q t + q^{2}) + h + k = 0$ $\Rightarrow 2 t^{3} + 3 (2 q - h) t^{2} + 6 q (q - h) t$ $+ 2 q^{3} - 3 {h q}^{2} + h + k = 0$ $n o w i f$ $9 {(2 q - h)}^{2} = 36 q (q - h)$ $\Rightarrow {(\frac{q}{h} - \frac{1}{2})}^{2} = (\frac{q}{h} - \frac{1}{2}) - \frac{1}{2}$ $\Rightarrow \frac{q}{h} - \frac{1}{2} = \frac{1}{2} \pm \sqrt{\frac{1}{4} - \frac{1}{2}}$ $T h i s m u s t b e q u a d r a t i c$ $\Rightarrow h + k = 0$ $h = \frac{2 s}{3}$ $N o w s i n c e$ $m = 3 s^{2} - 1 = \frac{p - h}{k}$ $\Rightarrow 3 s^{2} - 1 = - \frac{p}{h} + 1$ $p = (\frac{2 s}{3}) (2 - 3 s^{2})$ $p^{3} = p + c$ $O t h e r w i s e e q . o f n o r m a l i s$ $y - k = - \frac{(x - h)}{(3 s^{2} - 1)}$ $y + \frac{2 s}{3} + \frac{x - \frac{2 s}{3}}{3 s^{2} - 1} = 0$ $(p, 0) l i e s o n i t; h e n c e$ $p = \frac{2 s}{3} (2 - 3 s^{2})$
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