Question

For the Hyperbola `x^2/(cos^2α) - y^2/(sin^2α)` = 1, which of the following remains constant when α varies = ?

Options

• abscissae of vertices

• abscissae of foci

• eccentricity

• directrix

Answer 1

Abscissae of foci

Explanation:

Given, equation of the hyperbola is

`x^2/a^2 - y^2/b^2` = 1, we get a² = cos² α and b² = sin² α

We know that, b² = a²(e² – 1)

⇒ sin² α = cos² α(e² – 1)

⇒ sin² α + cos² α = cos² α.e²

⇒ e² = sec² α

⇒ e = sec α

∴ ae = `cos α. 1/(cosα)` = 1

Coordinates of foci are (±ae, 0) i.e. (± 1, 0)

Hence, the abscissae of foci remain constant when α varies.