\(\text{Curve is an ellipse} \ \ \Rightarrow \ \ \text {Eliminate B}\)

\(\text{Trig identity:}\)

\(\sin ^2 \theta+\cos ^2 \theta=1 \ \Rightarrow \ \tan ^2 \theta+1=\sec ^2 \theta \ \ \text{(Divide by \(\cos ^2 \theta\))} \)
 

\(\text{Consider option A:}\)

\(x=2 \sec \theta-1 \ \Rightarrow \ \sec \theta=\dfrac{x+1}{2}\)

\(y=3 \tan \theta+2 \ \Rightarrow \ \tan \theta=\dfrac{y-2}{3}\)

\(\dfrac{(y-2)^2}{9}+1=\dfrac{(x+1)^2}{4} \quad\)X

 
\(\text{Consider option C:}\)

\(x=2 \tan \theta-1 \ \Rightarrow \ \tan \theta=\dfrac{x+1}{2}\)

\(y=3 \sec \theta+2 \ \Rightarrow \ \sec \theta=\dfrac{y-2}{3}\)

\(\dfrac{(x+1)^2}{4}+1=\dfrac{(y-2)^2}{9} \quad \large{\checkmark}\)

\(\Rightarrow C\)