Question

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6. The equation of the hyperbola referred to its axes as axes of coordinates are:

• A. \( 3x^2 - y^2 = 3 \)
• B. \( x^2 - 3y^2 = 3 \)
• C. \( 3x^2 - y^2 = 9 \)
• D. \( x^2 - 3y^2 = 9 \)

Hint:

For hyperbolas, use the relationship \( c^2 = a^2 + b^2 \) to find the parameters of the equation and apply known conditions like conjugate axis length.

Answer 1

The Correct Option is C

Step 1: General form of a hyperbola.
The standard form of a hyperbola with a horizontal transverse axis is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The distance between the foci is \( 2c \), and the distance between the vertices is \( 2a \). We are told that the distance between the foci is double the distance between the vertices, so: \[ 2c = 2 \times 2a \quad \implies \quad c = 2a \]

Step 2: Relationship between \( a \), \( b \), and \( c \).
For hyperbolas, we know that: \[ c^2 = a^2 + b^2 \] Substitute \( c = 2a \) into this equation: \[ (2a)^2 = a^2 + b^2 \quad \implies \quad 4a^2 = a^2 + b^2 \] Simplifying: \[ b^2 = 3a^2 \]

Step 3: Use the given conjugate axis length.
We are told that the length of the conjugate axis is 6, so: \[ 2b = 6 \quad \implies \quad b = 3 \] Thus, \( b^2 = 9 \).

Step 4: Substitute into the equation of the hyperbola.
Substitute \( a^2 \) and \( b^2 \) into the standard form of the hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{9} = 1 \] From \( b^2 = 3a^2 \), we find that \( a^2 = 3 \), so the equation becomes: \[ \frac{x^2}{3} - \frac{y^2}{9} = 1 \] Multiplying both sides by 9: \[ 3x^2 - y^2 = 9 \]

Step 5: Conclusion.
Thus, the equation of the hyperbola is \( 3x^2 - y^2 = 9 \), corresponding to option (C).