Question

If the equation \( \frac{x^2}{7-k} - \frac{y^2}{5-k} = 1 \) represents a hyperbola, then:

• A. \( 5<k<7 \)
• B. \( k<5 \) or \( k>7 \)
• C. \( k>5 \)
• D. \( k \neq 5, k \neq 7, -\infty<k<\infty \)

Hint:

For an equation of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) to represent a hyperbola, the denominator of \( x^2 \) must be positive, and the denominator of \( y^2 \) must be negative.

Answer 1

The Correct Option is A

Step 1: Identify the Given Equation Type
The general equation of a hyperbola is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] Comparing with the given equation: \[ \frac{x^2}{7-k} - \frac{y^2}{5-k} = 1, \] we identify: \[ a^2 = 7 - k, \quad b^2 = 5 - k. \] Step 2: Condition for a Hyperbola
For the equation to represent a hyperbola, the denominator of \( x^2 \)
(i.e., \( 7 - k \)) must be positive, and the denominator of \( y^2 \)
(i.e., \( 5 - k \)) must be negative:
\[ 7 - k>0, \quad 5 - k<0. \] Solving these inequalities: 1. \( 7 - k>0 \) \[ k<7. \] 2. \( 5 - k<0 \) \[ k>5. \] Step 3: Conclusion
From the inequalities: \[ 5<k<7. \] Thus, the correct answer is: \[ \mathbf{5<k<7}. \]