Write the equations of the tangent and normal

Write the equations of the tangent and normal to the curve $x=\dfrac{2t}{t+2},\ y=\dfrac{t}{t-1}$ at the point $(1,2)$.

Step 1: Find the parameter at $(1,2)$

\[ \frac{2t}{t+2}=1 \;\Longrightarrow\; 2t=t+2 \;\Longrightarrow\; t=2, \qquad y(2)=\frac{2}{1}=2.\ \checkmark \]

Step 2: Parametric derivatives

\[ \dot x = \frac{2(t+2)-2t}{(t+2)^{2}}=\frac{4}{(t+2)^{2}}, \qquad \dot y = \frac{(t-1)-t}{(t-1)^{2}}=\frac{-1}{(t-1)^{2}}. \tag{1} \]

Step 3: Slope at $t=2$

\[ \dot x(2)=\frac{4}{16}=\frac14, \qquad \dot y(2)=-1, \qquad k_{t}=\frac{-1}{1/4}=-4, \qquad k_{n}=\frac14. \]

Step 4: Verification

\[\text{Both lines pass through } (1,2)\text{: } -4(1)+6=2\ \checkmark;\quad \tfrac14(1)+\tfrac74=2\ \checkmark\]

\[\text{Slopes perpendicular: } (-4)\cdot\tfrac14=-1\ \checkmark\]