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Find the eccentricity of the conic defined by the equation (2x - 4)² + 4y² = (x + y + 1)².

1. 1/2
2. 1/sqrt(2)
3. 2
4. sqrt(2)

Correct answer: 1/sqrt(2)

Solution

Rewriting gives 4[(x-2)²+y²] = (x+y+1)². The left side is [2*d_focus]² and the right side is [sqrt(2)*d_line]², so d_focus = (1/sqrt(2))*d_line, meaning the eccentricity e = 1/sqrt(2) < 1, which is an ellipse.

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