Question

Vă rog, problema 75.

Answer 1

$A(-3,-1),\quad B\Big(2,\dfrac{11}{4}\Big),\quad Q\Big(3,\dfrac{1}{2}\Big) \\ \ \begin{array}{rcl}\overrightarrow{AB} &=& (2-(-3)\vec{i} +\Big(\dfrac{11}{4}-(-1)\Big)j \\ &=& 5\vec{i} +\dfrac{15}{4}\vec{j} \end{array} \\ \\ \\ AB || DC \Rightarrow \left|\begin{array}{rcl}\overrightarrow{DC} &=& \overrightarrow{AB} \\ &=& 5\vec{i} +\dfrac{15}{4}\vec{j} \end{array}\right|$

$\text{Il aflam pe } D(d_1,d_2) }:\\ \\ BQ = QD \Rightarrow Q\Big(\dfrac{x_B+x_D}{2},\dfrac{y_B+y_D}{2}\Big) \Rightarrow \\ \\ \Rightarrow Q\Big(\dfrac{2+d_1}{2},\dfrac{\frac{11}{4}+d_2}{2}\Big) \Rightarrow \\ \\ \dfrac{2+d_1}{2} = 3 \Rightarrow d_1 = 4\\ \\ \dfrac{\frac{11}{4} +d_2}{2} = \dfrac{1}{2} \Rightarrow d_2 = 1-\frac{11}{4} \Rightarrow d_2 = -\dfrac{7}{4}] \\ \\ \Rightarrow \boxed{D\Big(4,-\dfrac{7}{4}\Big)}$

$\text{Il aflam pe }C(c_1,c_2): \\ \\ \overrightarrow{DC} = (c_1-4)i + \Big(c_2 + \dfrac{7}{4}\Big)\vec{j} \\ \overrightarrow{DC} = 5\vec{i}+\dfrac{15}{4}\vec{j} \\ \\ c_1-4 = 5 \Rightarrow c_1 = 9 \\ c_2 +\dfrac{7}{4} = \dfrac{15}{4} \Rightarrow c_2 = 2\\ \\ \Rightarrow \boxed{C(9,2)}$

$A(-3,-1),\quad C(9,2). \\ \\ \left|\begin{array}{ccc}x&y&1\\x_A&y_A&1\\x_C&y_C&1\end{array}\right| = 0 \Rightarrow \left|\begin{array}{ccc}x&y&1\\-3&-1&1\\9&2&1\end{array}\right| = 0 \Rightarrow \\ \\ \Rightarrow -x-6+9y + 9 - 2x +3y = 0 \Rightarrow \\ \\ \Rightarrow AC: -3x+12y+3 = 0\Big|:(-3) \Rightarrow \boxed{AC: x-4y -1 = 0}$