The angle between the asymptotes of (x^2)/(a^2)-(y^2)/(b^2)=1 is equal to (a) tan^-1 a (b) tan^-

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The angle between the asymptotes of...

The angle between the asymptotes of $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is equal to
(a) $\tan ^{-1} \mathrm{a}$
(b) $\tan ^{-1} \mathrm{~b}$
(c) $\tan ^{-1} \frac{\mathrm{b}}{\mathrm{a}}$
(d) $2 \tan ^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

Key Concepts

Hyperbola

A hyperbola is a type of conic section defined by a second-degree equation in two variables, typically representing the locus of points where the absolute difference of the distances from two fixed points (the foci) is constant. It has two separate branches and is often studied in relation to other conic sections such as ellipses and parabolas.

Asymptotes of a Hyperbola

Asymptotes are straight lines that the branches of a hyperbola approach but never touch as they extend to infinity. For a hyperbola in standard form, the equations of the asymptotes can be derived from the equation of the hyperbola by setting the constant term to zero, yielding the lines that define the general slant of the hyperbola.

Angle Between Two Lines

The angle between two lines can be determined using the slopes of the lines. When the slopes are known, the formula for the angle between the lines involves the difference of the slopes. In the special case of lines that are symmetric with respect to an axis, such as the asymptotes of a hyperbola, the angle between them can be calculated using a simplified inverse tangent expression.

Inverse Tangent (Arctan) Function

The inverse tangent function, denoted as tan?¹, returns the angle whose tangent is a given number. It is commonly used in trigonometry to determine angles when the ratio of the lengths of the opposite side to the adjacent side in a right triangle is known, and it plays a crucial role in computing the angle between lines given their slopes.

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