Find the length of the latus rectum of the hyperbola x² - y² =  1 ?

CONCEPT:

The properties of a rectangular hyperbola \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) are:

• Its centre is given by: (0, 0)
• Its foci are given by: (- ae, 0) and (ae, 0)
• Its vertices are given by: (- a, 0)  and (a, 0)
• Its eccentricity is given by: \(e = \frac{{\sqrt {{a^2} + {b^2}} }}{a}\)
• Length of transverse axis = 2a and its equation is y = 0.
• Length of conjugate axis = 2b and its equation is x = 0.
• Length of its latus rectum is given by: \(\frac{2b^2}{a}\)

CALCULATION:

Given: Equation of hyperbola is x2 - y2 =  1

As we can see that, the given hyperbola is a horizontal hyperbola.

So, by comparing the given equation of hyperbola with \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) we get

⇒ a = 1 and b = 1

As we know that, length of latus rectum of a hyperbola is given by \(\frac{2b^2}{a}\)

So, the length of latus rectum of given hyperbola is 2 units.

Hence, option D is the correct answer.