Conic Sections

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Step 1: A plane perpendicular to the axis and cutting one nappe completely forms a circle. Step 2: For a circle, eccentricity is 0. Answer: Circle.

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Step 1: Cutting one nappe completely without being perpendicular to the axis gives an ellipse. Step 2: An ellipse has eccentricity less than 1. Answer: Ellipse.

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Step 1: A plane parallel to the side of the cone produces a parabola. Step 2: A parabola has eccentricity 1. Answer: Parabola.

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Step 1: Cutting across both nappes gives a hyperbola. Step 2: A hyperbola has eccentricity greater than 1. Answer: Hyperbola.

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Step 1: Write it as ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Here, a = 1, 2h = 4 so h = 2, and b = 4. Step 2: Check h^2 and ab. h^2 = 2^2 = 4 and ab = 1×4 = 4. Step 3: Since h^2 = ab and Δ is not zero,…

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Step 1: Compare with ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Here, a = 1, b = 1, h = 0. Step 2: For a circle, Δ must not be 0, a = b, and h = 0. Step 3: Since a = b = 1 and h = 0, check Δ. Δ = 0 for t…

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Step 1: Compare with ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Here, a = 9, h = 0, b = -4. Step 2: Compute h^2 and ab. h^2 = 0 and ab = 9×(-4) = -36. Step 3: A hyperbola needs h^2 > ab and Δ not equal t…

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Step 1: For a parabola, the distance from any point to the focus is equal to its perpendicular distance from the directrix. Step 2: Eccentricity is the ratio of these two distances. Step 3: Since the …