Step 1 we have to eliminate the parameter θ in the following parametric equations: \(x=a\ tan\ \theta\)

\(y=b\sec\ \theta\)

as \(x=a\ \tan \theta\) therefore, \(\tan \theta = \frac{x}{a}\) (1) as \(y=b\sec \theta\) therefore, \(\sec \theta =\frac{y}{b}\) (2) Step 2 as we know that: \(\sec^2 \theta − \tan^2 \theta = 1\) therefore, \((\sec \theta)^2 − (\tan \theta)^2\) \((\frac{y}{b})2 − (\frac{x}{a})^2 = 1\) (from equation (1) and (2)) \(\frac{y^2}{b^2} − \frac{x^2}{a^2} = 1\) therefore the equation of the hyperbola obtained after eliminating the parameter \(\theta\) in the parametric equations is: \(\frac{y^2}{b^2} − \frac{x^2}{a^2} = 1\)