Maths
Comment déterminer l'équation réduite d'une droite connaissant les coordonnées de deux de ses points.
Comment déterminer l'équation réduite d'une droite connaissant deux de ses points.
Le problème.
Dans un repère, on connait les coordonnées de deux points appartenant à une droite. On cherche à déterminer une équation de la doite.
Les propriétés.
Soit les coordonnées d'un point A du plan et y = mx + p l'équation réduite d'une droite (d) non parallèle à l'axe des ordonnées. Dire que A
(d) est équivalent à dire
.
Autrement dit, que les abscisses et ordonnées du point vérifient l'équation de la droite.
On peut aussi se servir du coefficient directeur m :
Si le coefficient directeur n'existe pas, alors la droite est parallèle à l'axe des ordonnées. ATTENTION ! Si m = 0, le coefficient directeur existe !!! La droite est alors parallèle à l'axe des abscisses.
Les méthodes.
On peut procéder de deux manières :
Comme on a deux points, on monte un système en égalisant les abscisses et les ordonnées des deux points dans l'équation de la droite. On obtient :
On résout ce système en m et p.
Ou bien, on se sert du coefficient directeur comme il est indiqué dans les propriétés au paragraphe précédent. Le calcul de l'ordonnée à l'origine est trivial.
L'exemple.
On veut connaitre l'équation réduite de la droite qui passe Par les points A(-1 ; 2) et B(1 ; -1)
Premiere méthode, on remplace x et y par respectivement les abscisses et les ordonnées des points et on obtient le système suivant :
Soit 2p = 1 et et
.
L'équation réduite de la droite passant par A et B est
Deuxième méthode, en se servant de la formule du coefficient directeur.
On a
En remplaçant par les coordonnées des deux points.
On réécrit l'équation en prenant les coordonnées du point A (s'aurait pu être celles de B) soit :
et
.
On a notre équation.
Last update: 2006-02-08 18:55
Author: Henry Serge
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