Question

Find the equation of the hyperbola, the length of whose latus rectum is 4  and the eccentricity is 3 ?
1. 4x² - 2y² = 1
2. 16x² - 2y² = 1
3. 4x² - 3y² = 1
4. 3x² - 4y² = 1

Answer 1

Correct Answer - Option 2 : 16x² - 2y² = 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:

Here, we have to find the equation of hyperbola whose length of latus rectum is 4 and the eccentricity is 3.

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

⇒ \(\frac{2b^2}{a} = 4\)

⇒ b2 = 2a

As we know that, the eccentricity of a hyperbola is given by \(e = \frac{{\sqrt {{a^2} + {b^2}} }}{a}\)

⇒ a2e2 = a2 + b2

⇒ 9a2 = a2 + 2a

⇒ 8a² = 2a

⇒ a = 1/4 

⇒ a² = 1/16

∵ b2 = 2a

⇒ b2 = 1/2

So, the equation of the required hyperbola is 16x2 - 2y2 = 1

Hence, option B is the correct answer.