Question

Distance between two foci of the hyperbola \( x^2 - 4y^2 = 16 \) is

• A. 4
• B. 6
• C. 8
• D. 10

Hint:

For a hyperbola, the distance between the foci is \( 2 \times \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.

Answer 1

The Correct Option is C

The equation of the hyperbola is \( x^2 - 4y^2 = 16 \). This is of the standard form: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Comparing this with the given equation, we see that: \[ a^2 = 16 \quad \text{and} \quad b^2 = 4 \] Thus, \( a = 4 \) and \( b = 2 \). The distance between the foci of a hyperbola is given by \( 2c \), where \( c = \sqrt{a^2 + b^2} \). Substituting the values of \( a \) and \( b \): \[ c = \sqrt{a^2 + b^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \] Therefore, the distance between the foci is: \[ 2c = 2 \times 2\sqrt{5} = 8 \] So, the correct answer is 8.