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A circle through S, V and B also passes through J. Conversely, if a point, B on the parabola VG is to be found so that the area of the sector SVB is equal to a specified value, determine the point J on VX and construct a circle through S, V and J.
Any curve x(u 0,v) with fixed parameter ... If the directrix is a circle, the conoid is called a circular conoid. ... describes a parabolic conoid with the equation = ...
In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies on S. Examples include the plane , the lateral surface of a cylinder or cone , a conical surface with elliptical directrix , the right conoid , the helicoid , and the tangent developable of a ...
The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at (x 0, y 0) orthogonally, the following equations hold: = = The last equation is equivalent to =, which is the equation of the directrix.
More generally, for any collection of points P i, weights w i, and constant C, one can define a circle as the locus of points X such that (,) =.. The director circle of an ellipse is a special case of this more general construction with two points P 1 and P 2 at the foci of the ellipse, weights w 1 = w 2 = 1, and C equal to the square of the major axis of the ellipse.
A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone.
If you've been having trouble with any of the connections or words in Thursday's puzzle, you're not alone and these hints should definitely help you out. Plus, I'll reveal the answers further down ...
Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the ...