Question

If \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents a joint equation of directrices of the hyperbola \( 16x^2 - 9y^2 = 144 \), then \( g + f - c = \)

• A. -81
• B. -25
• C. 81
• D. 25

Hint:

For conic sections, the relationship between the equation of the conic and the coefficients of the directrix can help simplify the problem.

Answer 1

The Correct Option is C

Step 1: Write the equation of the hyperbola.
The standard form of the hyperbola is given as: \[ 16x^2 - 9y^2 = 144 \quad \Rightarrow \quad \frac{x^2}{9} - \frac{y^2}{16} = 1 \] This represents the equation of a hyperbola with center at the origin, and the coefficients for the directrix can be found using this standard equation.
Step 2: Identify the coefficients.
Comparing the equation of the hyperbola with the general form, we can determine the values of \( g \), \( f \), and \( c \). After simplifying, we find that \( g + f - c = 81 \).
Step 3: Conclusion.
The correct answer is (C) 81.