Question

The distance between the foci of a hyperbola is 16 and its eccentricity is ${\displaystyle \sqrt{2}}$. Its equation is ______.

पर्याय

• x² – y² = 32

• ${\displaystyle \frac{x^2}{4}-\frac{y^2}{9}}$ = 1

• 2x² – 3y² = 7

• None of these

Answer 1

The distance between the foci of a hyperbola is 16 and its eccentricity is ${\displaystyle \sqrt{2}}$. Its equation is x² – y² = 32.

Explanation:

We know that the distance between the foci = 2ae

∴ 2ae = 16

⇒ ae = 8

Given that e = ${\displaystyle \sqrt{2}}$

∴ ${\displaystyle \sqrt{2}a}$ = 8

⇒ ${\displaystyle a=4\sqrt{2}}$

Now b² = a² (e²  – 1)

⇒ b² = 32(2 – 1)

⇒ b² = 32

So, the equation of the hyperbola is ${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}}$ = 1

⇒ ${\displaystyle \frac{x^2}{32}-\frac{y^2}{32}}$ = 1

⇒ x² – y² = 32