Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
Equations of the asymptotes of the hyperbola whose equation is given by $x = a \tan(\theta + \alpha)$ and $y = b \tan(\theta + \beta)$, $\theta$ being a parameter, is/are:
$by = ax \tan(\alpha - \beta)$
$y = ax \tan(\alpha - \beta)$
$x + a \cot(\alpha - \beta) = 0$
$y - b \cot(\alpha - \beta) = 0$
Step-by-Step Solution
Key Concept: Convert parametric equations using tan difference formula: tan(α - β) = [tan(α) - tan(β)]/[1 + tan(α)tan(β)], then substitute x = a·tan(θ + α) and y = b·tan(θ + β) to derive the rectangular hyperbola equation xy - (bx - ay)cot(α - β) + ab = 0, whose asymptotes are found by equating the linear terms to zero.
For parametric angles $\alpha$ and $\beta$ on a rectangular hyperbola, $\tan(\alpha - \beta) = \frac{\frac{x}{a} - \frac{y}{b}}{1 + \frac{x}{a}\cdot\frac{y}{b}}$. This simplifies to $xy - (bx - ay)\cot(\alpha - \beta) + ab = 0$, a rectangular hyperbola. The asymptotes are $x + a\cot(\alpha - \beta) = 0$ and $y - b\cot(\alpha - \beta) = 0$.
Correct Answer: 3,4