A section of a right circular cone by a plane perpendicular to its axis is
Answer: A
Cutting square across the axis gives a curve everywhere equidistant from it, which is the limiting case of the ellipse with eccentricity zero. Tilting the plane even slightly away from perpendicular begins at once to stretch it into an ellipse.
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A plane cutting a cone at an angle to the axis greater than the generator makes gives
Answer: B
Correct answer: an ellipse
Measured against the axis, the generator sets the dividing angle. A plane at a greater angle to the axis than the generator cuts right through the cone and closes, giving an ellipse; at exactly the generator’s angle it never closes and gives a parabola; and at a smaller angle it meets both nappes and gives a hyperbola. Perpendicular to the axis gives the circle.
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A plane parallel to a generator of the cone gives
Answer: C
Correct answer: a parabola
A plane parallel to a generator is at exactly the critical inclination: it cuts one nappe only, but never closes, because it runs alongside the very line it would have to cross to come back. The result is the single open branch of a parabola, and it is the boundary case between the ellipse and the hyperbola.
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To obtain a section with two separate branches, the cutting plane must be inclined to the axis
Answer: D
Correct answer: more gently than the generator is
Two branches require the plane to meet both nappes of the double cone, which happens when it is inclined to the axis more gently than the generator - that is, when it is steeper relative to the base. At exactly the generator’s inclination it touches one nappe only and gives a parabola.
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A plane passing through the apex of a cone gives, in the general case,
Answer: A
Correct answer: two intersecting straight lines
A plane through the apex meets the surface in generators rather than in a curve. In the general case it cuts two of them, giving a pair of straight lines crossing at the apex; tangential to the cone it gives one; and passing through the apex without entering the cone, only the apex itself. These are the degenerate conics.
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The eccentricity of a conic is the ratio of the distance of a point from the focus to its distance from the
Answer: B
That single ratio distinguishes all four conics from one another and is constant for every point on a given curve. It is the basis of the eccentricity method of construction, which draws any conic from a focus and a directrix alone.
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The eccentricity of an ellipse is
Answer: C
Correct answer: less than one
Eccentricity is the ratio of the distance from a point on the curve to the focus and to the directrix, and it classifies the conics completely: zero for a circle, between zero and one for an ellipse, exactly one for a parabola, and greater than one for a hyperbola. The nearer to zero, the more nearly circular the ellipse.
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The eccentricity of a parabola is
Answer: D
Every point is equidistant from focus and directrix, which is the defining property and the reason the curve neither closes on itself nor divides into branches. It marks the exact boundary between the ellipse and the hyperbola.
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A circle may be regarded as an ellipse whose eccentricity is
Answer: A
The two foci coincide at the centre and the directrix recedes to infinity, so the ratio vanishes altogether. It is why the circle is treated as the limiting member of the conic family rather than as a separate kind of curve.
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The latus rectum of a conic is the chord through the focus drawn
Answer: B
Correct answer: perpendicular to the axis
The latus rectum is the focal chord at right angles to the axis, and its length is a natural measure of how open the curve is. For a parabola it is four times the distance from vertex to focus, which is the quickest way of setting out the width of the curve once the focus is fixed.
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