Question

Sa se determine coordonatele proiectiilor punctului A(2, 3) pe dreapta de ecuatie: 3x+2y−1 = 0,

Answer 1

Luam un punct M proiectia lui A pe dreapta.

AM ⊥ d ⇔ $m_{AM} * m_{d} = -1$

$d:3x+2y-1=0 \\ \\ d:2y=1-3x \\ \\ d:y= -\frac{3x}{2} + \frac{1}{2} \\ \\ m_{d}=- \frac{3}{2}$

$m_{AM} *m_{d} =-1 \\ \\ m_{AM}* (-\frac{3}{2})=-1 \\ \\ m_{AM}= \frac{2}{3}$

$d_{AM}: y-y_{A}=m_{AM}(x- x_{A}) \\ \\ d_{AM}:y-3 = \frac{2}{3}(x-2) \\ \\ d_{AM}: 3y-9=2x-4 \\ \\ d_{AM}:2x-3y+5=0$

Ca sa aflam intersectia dreptelor, facem sistem de ecuatii.

$\left \{ {{3x+2y-1=0} \atop {2x-3y+5=0}} \right.~(inmultim~prima~ cu ~-2 ~si~ a~treia~ cu~3) \\ \\ \left \{ {-6x-4y+2=0} \atop {6x-9y+15=0}} \right. \\ \\ Le~adunam \\ \\ -13y=-17 \\ \\ y=\frac{17}{13} \\ \\ \\ 2x=3y-5 \\ \\ 2x= \frac{51}{13}-5 \\ \\ 2x= \frac{51-65}{13} \\ \\ 2x= -\frac{14}{13} \\ \\ x= -\frac{7}{13} \\ \\ \\ Raspuns:M( -\frac{7}{13} ; \frac{17}{13} )$