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If theta is eliminated from the equations a*sec(theta) - x*tan(theta) = y and b*sec(theta) + y*tan(theta) = x (where a and b are constants), the resulting equation represents:

1. the director circle of the hyperbola x²/a² - y²/b² = 1
2. the auxiliary circle of the ellipse x²/a² + y²/b² = 1
3. the director circle of the ellipse x²/a² + y²/b² = 1
4. the director circle of the circle x² + y² = (a² + b²)/2

Correct answer: the director circle of the hyperbola x²/a² - y²/b² = 1

Solution

Eliminating theta via sec²(theta) - tan²(theta) = 1 yields x² + y² = a² + b², which is the director circle of the hyperbola x²/a² - y²/b² = 1.

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