Question

# 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. $$x = a\ tan\ 0\ y = b\ sec$$ e

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: $$\frac{y^2}{b^2} − \frac{x^2}{a^2} = 1$$