Conic Sections and Engineering Curves Questions and Answers

Answer: C

Correct answer: focus, directrix and eccentricity

The eccentricity method constructs the conic directly from its definition as the locus of a point whose distances from the focus and the directrix stand in a fixed ratio. Because that definition covers all four conics, the same construction serves for every one of them, with only the value of the eccentricity changing.

Answer: D

Correct answer: the sum of distances to the two foci is constant

This is the focal definition of the ellipse, and the arcs of circles method uses it directly: divide the major axis at a series of points, and from each strike arcs of the two complementary radii from the two foci to locate points on the curve. It is the same property the gardener’s loop of string exploits.

Answer: A

Correct answer: two circles on the major and minor axes as diameters

Draw both circles about the common centre and a radius across them; the point where the vertical from the outer circle meets the horizontal from the inner lies on the ellipse. It is quick and exact, needing only a compass and a set square, and it gives as many points as radii are drawn.

Answer: B

Correct answer: an approximate ellipse made of four circular arcs

The four-centre method joins four circular arcs, two of small radius at the ends of the major axis and two of large radius at the ends of the minor. It is quick and can be struck with a compass, which is its whole merit, but a circular arc is not an ellipse and the figure is an approximation.

Answer: C

Correct answer: the half base and the height into equal parts

Divide the half base and the height into the same number of equal parts, join each division of the height to the vertex and draw a vertical from the corresponding division of the base; the intersections lie on the parabola. Repeating on the other side completes the curve, and the construction needs no focus or directrix.

Answer: D

Correct answer: as the envelope of a series of tangent lines

The tangent method draws no points on the curve at all: it constructs a family of lines each touching the parabola, and the curve appears as the boundary they envelope. It is the natural construction when the parabola is specified by two tangents and their points of contact, as in the profile of an arch.

Answer: A

Correct answer: at right angles to one another

A rectangular hyperbola is the one whose asymptotes are perpendicular, which makes its eccentricity exactly the square root of two. Referred to those asymptotes as axes its equation is simply that the product of the coordinates is constant, which is why it describes an isothermal expansion and other inverse relationships.

Answer: B

Correct answer: normal at that point

The normal is by definition perpendicular to the tangent, so constructing one gives the other. In practice the normal is often easier to find first - for an ellipse it bisects the angle between the focal radii, and for a parabola it can be located from the focus and directrix - and the tangent is then drawn at right angles to it.

Answer: C

The string stays taut and tangential throughout, so each element of the curve is perpendicular to the string at that instant. Gear teeth are given involute flanks because the profile transmits motion at a constant velocity ratio.

Answer: D

Correct answer: along a straight line

A cycloid is the path of a point on the rim of a circle rolling without slipping along a straight line. Rolling outside another circle gives an epicycloid and inside one a hypocycloid, so the three form a family distinguished only by what the circle rolls along.