Resolved: "Parametric Equations of a Hyperbola Eliminate the parameter..."

College
Calculus and Analysis
Integral Calculus
Parametric equations, polar coordinates, and vector-valued functions

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Answered question

2021-01-19

Parametric Equations of a Hyperbola Eliminate the parameter 0 in the following parametric equations. $x = a t a n 0 y = b s e c$ e

Answer & Explanation

Nathalie Redfern

Skilled2021-01-20Added 99 answers

Step 1 we have to eliminate the parameter θ in the following parametric equations: $x = a t a n θ$
$y = b \sec θ$
as $x = a \tan θ$ therefore, $\tan θ = \frac{x}{a}$ (1) as $y = b \sec θ$ therefore, $\sec θ = \frac{y}{b}$ (2) Step 2 as we know that: $\sec^{2} θ - \tan^{2} θ = 1$ therefore, $(\sec θ)^{2} - (\tan θ)^{2}$ $(\frac{y}{b}) 2 - (\frac{x}{a})^{2} = 1$ (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 $θ$ in the parametric equations is: $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$

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