## (De high school Toan 2023) In the plane with the coordinate system (Oxygen) Let two pts ({Delta _1}) and ({Delta _2}) have the equations respectively.

• Question:

In the plane with the coordinate system $$Oxy$$ Given two lines $${\Delta _1}$$ and $${\Delta _2}$$ have the equation: $$x – 2y + 1 = respectively: 0$$ and $$x – 2y + 4 = 0$$, point $$I\left( {2;1} \right).$$ Self-centred predicate $$I$$ ratio $$k$$ turn the line $${\Delta _1}$$ into $${\Delta _2}.$$ Find $$k.$$

Reference explanation:

We take the point $$A\left( {1;1} \right) \in {\Delta _1}.$$ Then
$$A’ = {V_{\left( {I,k} \right)}}\left( A \right) \Rightarrow \left\{ {\begin{array}{*{20}{c}}{ x’ = kx + \left( {1 – k} \right)a}\\{y’ = ky + \left( {1 – k} \right)b}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{x’ = k + \left( {1 – k} \right)2}\\{y’ = k + \left( { 1 – k} \right)1}\end{array}} \right \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{x’ = 2 – k}\\{ y’ = 1}\end{array}} \right.$$
Which $$A’ \in {\Delta _2} \Rightarrow x’ – 2y’ + 4 = 0 \Rightarrow 2 – k – 2.1 + 4 = 0 \Rightarrow k = 4.$$