Question

În reperul cartezian xOy se consideră punctele A(2,0) , B(0,3) si dreapta
d:x-y+1=0 . Determinați coordonatele punctului C care aparține dreptei d , pentru care CA=CB

Answer 1

$A(2,0),\quad B(0,3),\quad d:x-y+1=0\\ \\ C(c_1,c_2) \in d \Rightarrow c_1 - c_2 + 1 = 0 \Rightarrow c_1 = c_2-1 \\ \\ CA = CB \Rightarrow \sqrt{(c_1 - 2)^2 + (c_2-0)^2} = \sqrt{(c_1-0)^2+(c_2-3)^2} \Rightarrow \\ \\ \Rightarrow (c_1-2)^2 +c_2^2 = c_1^2 +(c_2-3)^2 \\ \\ \Rightarrow \Big((c_2-1)-2\Big)^2 + c_2^2 = (c_2-1)^2 + (c_2-3)^2 \\ \\ \Rightarrow (c_2-3)^2 + c_2^2 = (c_2-1)^2+(c_2-3)^2 \\ \\ \Rightarrow c_2^2 = (c_2-1)^2 \\ \\ \Rightarrow c_2^2 - (c_2-1)^2 = 0$

$\Rightarrow \Big(c_2-(c_2-1)\Big)\Big(c_2+(c_2-1)\Big) = 0 \\ \\ \Rightarrow 1\cdot (2c_2-1) = 0 \\ \\ \Rightarrow 2c_2 = 1 \Rightarrow c_2 = \dfrac{1}{2} \\ \\ \Rightarrow c_1 = \dfrac{1}{2} - 1 \Rightarrow c_1 = -\dfrac{1}{2} \\ \\ \Rightarrow \boxed{C\left(-\dfrac{1}{2},\dfrac{1}{2}\right)}$