Question

The distance between the foci of a hyperbola is 16 and its eccentricity is \(\sqrt2\). Its equation is

• A. 2x² - 3y² = 7
• B. x² - y² = 32
• C. y² - x² = 32
• D. \(\frac{x^2}{4}-\frac{y^2}{9}=1\)

Hint:

When working with hyperbolas, remember that the relationship between the foci (c), the semi-major axis (a), and the semi-minor axis (b) is given by the equation \( c^2 = a^2 + b^2 \). Additionally, eccentricity \( e \) is defined as \( e = \frac{c}{a} \). These formulas are essential for deriving the equation of the hyperbola and solving related problems.

Answer 1

The Correct Option is B

The correct answer is (B) :x² - y² = 32 .

Answer 2

The correct answer is: (B) \( x^2 - y^2 = 32 \).

We are given that the distance between the foci of a hyperbola is 16, and its eccentricity is \( \sqrt{2} \). We need to find the equation of the hyperbola.

For a hyperbola, the general equation is:

\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

We also know the following relationships for a hyperbola:

• The distance between the foci is 2c, so c = 8.
• The eccentricity e = \(\frac{c}{a}\), and we are given that e = \( \sqrt{2} \).

From the equation e = \(\frac{c}{a}\), we can substitute the values:

\( \sqrt{2} = \frac{8}{a} \)

Solving for a:

\( a = \frac{8}{\sqrt{2}} = 4\sqrt{2} \)

Next, we use the relationship c² = a² + b² to find b²:

\( 8^2 = (4\sqrt{2})^2 + b^2 \)

\( 64 = 32 + b^2 \)

\( b^2 = 32 \)

Therefore, the equation of the hyperbola is:

\( \frac{x^2}{(4\sqrt{2})^2} - \frac{y^2}{32} = 1 \)

This simplifies to:

\( x^2 - y^2 = 32 \)

Thus, the correct equation of the hyperbola is (B) \( x^2 - y^2 = 32 \).