a) Gọi \(A' = {T_{\overrightarrow v }}\left( A \right) \Rightarrow \left\{ \begin{array}{l}{x_{A'}} = {x_A} - 3 = 1 - 3 = - 2\\{y_{A'}} = {y_A} + 4 = - 2 + 4 = 2\end{array} \right. \Rightarrow A'\left( { - 2;2} \right)\). * Gọi \(M\left( {x;y} \right) \in \Delta \). Gọi \(M'\left( {x';y'} \right) = {T_{\overrightarrow v }}\left( M \right) \Rightarrow \left\{ \begin{array}{l}x' = x - 3\\y' = y + 4\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = x' + 3\\y = y' - 4\end{array} \right. \Rightarrow M\left( {x' + 3;y' - 4} \right)\). \(M \in \Delta \Rightarrow 3\left( {x' + 3} \right) - 4\left( {y' - 4} \right) + 1 = 0 \Leftrightarrow 3x' - 4y' + 26 = 0\). Gọi \(\Delta ' = {T_{\overrightarrow v }}\left( \Delta \right) \Rightarrow M' \in \Delta ' \Rightarrow \) Phương trình \(\Delta ':\,\,3x - 4y + 26 = 0\). * Gọi \(M\left( {x;y} \right) \in \left( C \right)\). Ta có \(\left( C \right):\,\,{x^2} + {y^2} - 2x + 4y - 3 = 0 \Leftrightarrow {\left( {x - 1} \right)^2} + {\left( {y + 2} \right)^2} = 8\). Gọi \[\]. \(M \in \left( C \right) \Rightarrow {\left( {x' + 3 - 1} \right)^2} + {\left( {y' - 4 + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' + 2} \right)^2} + {\left( {y' - 2} \right)^2} = 8\). Gọi \(\left( {C'} \right) = {T_{\overrightarrow v }}\left( C \right) \Rightarrow M' \in \left( {C'} \right) \Rightarrow \) Phương trình \(\left( {C'} \right):\,\,{\left( {x + 2} \right)^2} + {\left( {y - 2} \right)^2} = 8\). b) Gọi \(M\left( {x;y} \right),\,\,M'\left( {x';y'} \right) = {D_O}\left( M \right) \Rightarrow \left\{ \begin{array}{l}x' = - x\\y' = - y\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = - x'\\y = - y'\end{array} \right. \Rightarrow M\left( { - x'; - y'} \right)\) Gọi \(A' = {D_O}\left( A \right) \Rightarrow A'\left( { - 1;2} \right)\). \(M \in \Delta \Rightarrow 3\left( { - x'} \right) - 4\left( { - y'} \right) + 1 = 0 \Leftrightarrow - 3x' + 4y' + 1 = 0\). Gọi \(\Delta ' = {D_O}\left( \Delta \right) \Rightarrow \Delta ':\,\, - 3x + 4y + 1 = 0\). \(M \in \left( C \right) \Rightarrow {\left( { - x' - 1} \right)^2} + {\left( { - y' + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' + 1} \right)^2} + {\left( {y' - 2} \right)^2} = 8\). Gọi \(\left( {C'} \right) = {D_O}\left( C \right) \Rightarrow \left( {C'} \right):\,\,{\left( {x + 1} \right)^2} + {\left( {y - 2} \right)^2} = 8\). Gọi \(M\left( {x;y} \right),\,\,M'\left( {x';y'} \right) = {D_I}\left( M \right) \Rightarrow \left\{ \begin{array}{l}x' = 10 - x\\y' = - 12 - y\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = 10 - x'\\y = - 12 - y'\end{array} \right. \Rightarrow M\left( {10 - x'; - 12 - y'} \right)\) Gọi \(A' = {D_O}\left( A \right) \Rightarrow A'\left( {9; - 10} \right)\). \(M \in \Delta \Rightarrow 3\left( {10 - x'} \right) - 4\left( { - 12 - y'} \right) + 1 = 0 \Leftrightarrow - 3x' + 4y' + 79 = 0\). Gọi \(\Delta ' = {D_O}\left( \Delta \right) \Rightarrow \Delta ':\,\, - 3x + 4y + 79 = 0\). \(M \in \left( C \right) \Rightarrow {\left( {10 - x' - 1} \right)^2} + {\left( { - 12 - y' + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' - 9} \right)^2} + {\left( {y' + 10} \right)^2} = 8\). Gọi \(\left( {C'} \right) = {D_O}\left( C \right) \Rightarrow \left( {C'} \right):\,\,{\left( {x - 9} \right)^2} + {\left( {y + 10} \right)^2} = 8\). c) \(M'\left( {x';y'} \right) = {D_{Ox}}\left( {M\left( {x;y} \right)} \right) \Rightarrow \left\{ \begin{array}{l}x' = x\\y' = - y\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = x'\\y = - y'\end{array} \right. \Rightarrow M\left( {x'; - y'} \right)\). Gọi \(A' = {D_{Ox}}\left( A \right) \Rightarrow A'\left( {1;2} \right)\). \(\begin{array}{l}M \in \Delta \Rightarrow 3x' - 4\left( { - y'} \right) + 1 = 0 \Leftrightarrow 3x' + 4y' + 1 = 0\\ \Rightarrow \Delta ':\,\,3x + 4y + 1 = 0\end{array}\). \(\begin{array}{l}M \in \left( C \right) \Rightarrow {\left( {x' - 1} \right)^2} + {\left( { - y' + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' - 1} \right)^2} + {\left( {y' - 2} \right)^2} = 8\\ \Rightarrow \left( {C'} \right):\,\,{\left( {x - 1} \right)^2} + {\left( {y - 2} \right)^2} = 8\end{array}\) \(M'\left( {x';y'} \right) = {D_{Oy}}\left( {M\left( {x;y} \right)} \right) \Rightarrow \left\{ \begin{array}{l}x' = - x\\y' = y\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = - x'\\y = y'\end{array} \right. \Rightarrow M\left( { - x';y'} \right)\). Gọi \(A' = {D_{Ox}}\left( A \right) \Rightarrow A'\left( { - 1; - 2} \right)\). \(\begin{array}{l}M \in \Delta \Rightarrow - 3x' - 4y' + 1 = 0 \Leftrightarrow 3x' + 4y' - 1 = 0\\ \Rightarrow \Delta ':\,\,3x + 4y - 1 = 0\end{array}\). \(\begin{array}{l}M \in \left( C \right) \Rightarrow {\left( { - x' - 1} \right)^2} + {\left( {y' + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' + 1} \right)^2} + {\left( {y' + 2} \right)^2} = 8\\ \Rightarrow \left( {C'} \right):\,\,{\left( {x + 1} \right)^2} + {\left( {y + 2} \right)^2} = 8\end{array}\) d) \(M'\left( {x';y'} \right) = {Q_{\left( {O;{{90}^0}} \right)}}\left( {M\left( {x;y} \right)} \right) \Rightarrow \left\{ \begin{array}{l}x' = - y\\y' = x\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = y'\\y = - x'\end{array} \right. \Rightarrow M\left( {y'; - x'} \right)\). Gọi \(A' = {Q_{\left( {O;{{90}^0}} \right)}}\left( A \right) \Rightarrow A'\left( {2;1} \right)\). \(\begin{array}{l}M \in \Delta \Rightarrow 3y' + 4x' + 1 = 0 \Leftrightarrow 4x' + 3y' + 1 = 0\\ \Rightarrow \Delta ':\,\,4x + 3y + 1 = 0\end{array}\). \(\begin{array}{l}M \in \left( C \right) \Rightarrow {\left( {y' - 1} \right)^2} + {\left( { - x' + 2} \right)^2} = 8 \Leftrightarrow {\left( {x' - 2} \right)^2} + {\left( {y' - 1} \right)^2} = 8\\ \Rightarrow \left( {C'} \right):\,\,{\left( {x - 2} \right)^2} + {\left( {y + 1} \right)^2} = 8\end{array}\) \(\begin{array}{l}M'\left( {x';y'} \right) = {Q_{\left( {O;{{60}^0}} \right)}}\left( {M\left( {x;y} \right)} \right) \Rightarrow \left\{ \begin{array}{l}x' = \frac{1}{2}x - \frac{{\sqrt 3 }}{2}y\\y' = \frac{{\sqrt 3 }}{2}x + \frac{1}{2}y\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = \frac{1}{2}x' + \frac{{\sqrt 3 }}{2}y'\\y = - \frac{{\sqrt 3 }}{2}x' + \frac{1}{2}y'\end{array} \right.\\ \Rightarrow M\left( {\frac{1}{2}x' + \frac{{\sqrt 3 }}{2}y'; - \frac{{\sqrt 3 }}{2}x' + \frac{1}{2}y'} \right)\end{array}\). Gọi \(A' = {Q_{\left( {O;{{60}^0}} \right)}}\left( A \right) \Rightarrow A'\left( {\frac{{1 + 2\sqrt 3 }}{2};\frac{{ - 2 + \sqrt 3 }}{2}} \right)\). \[\begin{array}{l}M \in \Delta \Rightarrow 3\left( {\frac{1}{2}x' + \frac{{\sqrt 3 }}{2}y'} \right) - 4\left( { - \frac{{\sqrt 3 }}{2}x' + \frac{1}{2}y'} \right) + 1 = 0\\ \Leftrightarrow 3x' + 3\sqrt 3 y' + 4\sqrt 3 x' - 4y' + 2 = 0\\ \Leftrightarrow \left( {3 + 4\sqrt 3 } \right)x' + \left( {3\sqrt 3 - 4} \right)y' + 2 = 0\\ \Rightarrow \Delta ':\,\,\,\left( {3 + 4\sqrt 3 } \right)x + \left( {3\sqrt 3 - 4} \right)y + 2 = 0\end{array}\]. \[\begin{array}{l}M \in \left( C \right) \Rightarrow {\left( {\frac{1}{2}x' + \frac{{\sqrt 3 }}{2}y' - 1} \right)^2} + {\left( { - \frac{{\sqrt 3 }}{2}x' + \frac{1}{2}y' + 2} \right)^2} = 8\\ \Rightarrow \left( {C'} \right):\,\,{\left( {\frac{1}{2}x + \frac{{\sqrt 3 }}{2}y - 1} \right)^2} + {\left( { - \frac{{\sqrt 3 }}{2}x + \frac{1}{2}y + 2} \right)^2} = 8\end{array}\] e) \(\begin{array}{l}M'\left( {x';y'} \right) = {V_{\left( {I; - 3} \right)}}\left( {M\left( {x;y} \right)} \right) \Rightarrow \overrightarrow {IM'} = - 3\overrightarrow {IM} \\ \Rightarrow \left\{ \begin{array}{l}x' - 5 = - 3\left( {x - 5} \right)\\y' + 6 = - 3\left( {y + 6} \right)\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x' = - 3x + 20\\y' = - 3y - 24\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = \frac{{20 - x'}}{3}\\y = \frac{{ - 24 - y'}}{3}\end{array} \right.\\ \Rightarrow M\left( {\frac{{20 - x'}}{3};\frac{{ - 24 - y'}}{3}} \right)\end{array}\) \(A' = {V_{\left( {I; - 3} \right)}}\left( A \right) \Rightarrow A'\left( {17; - 18} \right)\) \(\begin{array}{l}M \in \Delta \Rightarrow 3\frac{{20 - x'}}{3} - 4\frac{{ - 24 - y'}}{3} + 1 = 0\\ \Leftrightarrow 60 - 3x' + 96 + 4y' + 3 = 0 \Leftrightarrow - 3x' + 4y' + 159 = 0\\ \Rightarrow \Delta ':\,\, - 3x + 4y + 159 = 0\\M \in \left( C \right) \Rightarrow {\left( {\frac{{20 - x'}}{3} - 1} \right)^2} + {\left( {\frac{{ - 24 - y'}}{3} + 2} \right)^2} = 8\\ \Leftrightarrow {\left( {17 - x'} \right)^2} + {\left( { - 18 - y'} \right)^2} = 72\\ \Rightarrow \left( {C'} \right):\,\,{\left( {x - 17} \right)^2} + {\left( {y + 18} \right)^2} = 72\end{array}\)