Question

The equation of the conic with focus at \( (1, -1) \), directrix along \( x - y + 1 = 0 \), and eccentricity \( \sqrt{2} \) is:

• A. \( x^2 - y^2 = 1 \)
• B. \( xy = 1 \)
• C. \( 2xy - 4x + 4y + 1 = 0 \)
• D. \( 2xy + 4x - 4y - 1 = 0 \)

Hint:

For conics with a given eccentricity, focus, and directrix, use the relationship between the distance from a point on the conic to the focus and the distance from the point to the directrix to derive the equation.

Answer 1

The Correct Option is C

We are given the focus \( F(1, -1) \), the directrix \( x - y + 1 = 0 \), and the eccentricity \( e = \sqrt{2} \) of the conic. We need to find its equation. Step 1: General equation of the conic. The general equation of a conic with a focus \( F(x_1, y_1) \), directrix \( ax + by + c = 0 \), and eccentricity \( e \) is given by the formula: \[ e = \frac{r}{d}, \] where \( r \) is the distance from a point on the conic to the focus, and \( d \) is the distance from the point to the directrix. The equation of the conic can be derived from the definition of eccentricity, and for conics like the ellipse or hyperbola, we use the standard form \( r = e \cdot d \). Step 2: Equation for the given conic. For the given conic, since the focus is \( F(1, -1) \) and the eccentricity \( e = \sqrt{2} \), we use the following approach: 1. The equation of the directrix is \( x - y + 1 = 0 \). 2. The distance from any point \( (x, y) \) to the focus \( (1, -1) \) is given by: \[ r = \sqrt{(x - 1)^2 + (y + 1)^2}. \] 3. The distance from any point \( (x, y) \) to the directrix \( x - y + 1 = 0 \) is given by: \[ d = \frac{|x - y + 1|}{\sqrt{2}}. \] Thus, the equation becomes: \[ \sqrt{(x - 1)^2 + (y + 1)^2} = \sqrt{2} \cdot \frac{|x - y + 1|}{\sqrt{2}}. \] Simplifying: \[ \sqrt{(x - 1)^2 + (y + 1)^2} = |x - y + 1|. \] Step 3: Expanding and simplifying the equation. Squaring both sides: \[ (x - 1)^2 + (y + 1)^2 = (x - y + 1)^2. \] Expanding both sides: \[ (x^2 - 2x + 1) + (y^2 + 2y + 1) = (x^2 - 2xy + y^2 + 2x - 2y + 1). \] Simplifying: \[ x^2 - 2x + 1 + y^2 + 2y + 1 = x^2 - 2xy + y^2 + 2x - 2y + 1. \] Canceling common terms: \[ -2x + 1 + 2y + 1 = -2xy + 2x - 2y + 1. \] Simplifying further: \[ -2x + 2y + 2 = -2xy + 2x - 2y + 1. \] Now collect terms: \[ 2xy - 4x + 4y + 1 = 0. \] Step 4: Conclusion. Thus, the equation of the conic is \( 2xy - 4x + 4y + 1 = 0 \), and the correct answer is (c).