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Maths · Sections of conics, equations of conic sections (parabola, ellipse and hyperbola) in standard forms

For hyperbola which of the following remains constant with change in 'a'?

For hyperbola \( \frac{x^{2}}{\cos ^{2} a}-\frac{y^{2}}{\sin ^{2} a}=1 \) which of the following remains constant with change in 'a'?

  • A. Abscissae of vertices
  • B. Abscissae of foci
  • C. Eccentricity
  • D. Directrix

Step-by-step solution

The hyperbola is x^2/cos^2 a - y^2/sin^2 a = 1. Here A = |cos a|, B = |sin a|. For hyperbola, c^2 = A^2 + B^2 = cos^2 a + sin^2 a = 1, so foci are at (±1, 0). Thus abscissae of foci are constant (±1). Vertices (±cos a, 0), eccentricity e = 1/|cos a|, directrix x = ±cos^2 a all vary with a.
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