Question

Let \( (1, 2) \) be the focus and \( x + y + 1 = 0 \) be the directrix of a hyperbola. If \( \sqrt{3} \) is the eccentricity of the hyperbola, then its equation is:

• A. \( x^2 - 6xy + y^2 - 14x - 22y + 17 = 0 \)
• B. \( x^2 - 6xy + y^2 + 10x + 14y - 7 = 0 \)
• C. \( x^2 + 6xy + y^2 - 14x - 22y + 17 = 0 \)
• D. \( x^2 + 6xy + y^2 - 14x + 14y - 7 = 0 \)

Hint:

For hyperbolas, use the relationship between the focus, directrix, and eccentricity to derive the equation. The eccentricity gives the scaling factor in the formula.

Answer 1

The Correct Option is D

We are given the focus \( (1, 2) \), the directrix \( x + y + 1 = 0 \), and the eccentricity \( e = \sqrt{3} \). Step 1: The equation of a hyperbola with focus \( (x_1, y_1) \) and directrix \( ax + by + c = 0 \) is given by the formula: \[ \frac{(x - x_1)^2 + (y - y_1)^2}{(ax + by + c)^2} = e^2 \] Substitute the given values and simplify. Step 2: After simplification, the equation of the hyperbola becomes: \[ x^2 + 6xy + y^2 - 14x + 14y - 7 = 0 \] % Final Answer The equation of the hyperbola is \( x^2 + 6xy + y^2 - 14x + 14y - 7 = 0 \).