# Parametric Equations of a Hyperbola Eliminate the parameter 0 in the following parametric equations. x = a tan 0 y = b sec e

Parametric Equations of a Hyperbola Eliminate the parameter 0 in the following parametric equations. e

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Step 1 we have to eliminate the parameter θ in the following parametric equations:

as therefore, $\mathrm{tan}\theta =\frac{x}{a}$ (1) as $y=b\mathrm{sec}\theta$ therefore, $\mathrm{sec}\theta =\frac{y}{b}$ (2) Step 2 as we know that: ${\mathrm{sec}}^{2}\theta -{\mathrm{tan}}^{2}\theta =1$ therefore, $\left(\mathrm{sec}\theta {\right)}^{2}-\left(\mathrm{tan}\theta {\right)}^{2}$ $\left(\frac{y}{b}\right)2-\left(\frac{x}{a}{\right)}^{2}=1$ (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: $\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1$