Point of Beginning

Answers: Feb. 4

February 11, 2008
CORRECTED SOLUTION FOR PROBLEM  71(1-63)


What is the figure that will result from the cutting of a right circular cone by a plane parallel with the cone’s axis of symmetry?

(A) a loxodrome
(B) an ellipse
(C) a catenary
(D) a parabola

Answer is (D)

Solution: The figure that results is a parabola, also known as a conic section. Vertical curves are parabolic curves. This is problem 71(1-63) from the new second edition of 1001 Solved Surveying Problems by Jan Van Sickle. Reprinted with permission from 1001 Solved Surveying Problems by Jan Van Sickle (1997, 728 pp., Professional Publications Inc.). For details on this and other FLS exam-prep books, call 800/426-1178 or visit www.ppi2pass.com .

CORRECTED SOLUTION FOR PROBLEM 71(1-63)

POB extends our thanks to Troy J. Groth, PE, of Sundquist Engineering, PC, in Iowa, who spotted an error in this week's solution.

The following is the correct answer along with Mr. Van Sickle's explanation:
 
The correct answer is: a hyperbola
 
“A parabola is - open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone.”
In other words, if the plane that cuts the cone is parallel to one side of the cone a parabola is created.
  
However,
“When a slice is taken through a right circular cone such that its slope exceeds the slope of the cone, the perimeter of the slice defines an hyperbola.” http://www.space.gc.ca/asc/eng/educators/resources/orbital/geometry.asp
In other words, if the plane that cuts the cone is parallel to the axis of symmetry of the cone (as stated in the question) a hyperbola is created.
 
Specifically a parabola is the curve for which the distance from a fixed line (the directrix) is equal to its distance from a fixed point (the focus) whereas a hyperbola is the curve for which the difference between the distances to the foci remains constant.  Or stated another way the arms of a parabola eventually become parallel to each other, while the arms of a hyperbola always make an angle relative to each other
 
Some useful illustrations of the differences are available here.