Proof (a) Prove that if any two tangent lines to a parabola intersect at right angles, then their point of intersection must lie on the directrix. (b) Demonstrate the result of pan (a) by showing that the tangent lines to the parabola x 2 − 4 x − 4 y + 8 = 0 at the points ( − 2 , 5 ) and ( 3 , 5 4 ) intersect at right angles and that their point of intersection lies on the directrix.
Solution Summary: The author explains that if two tangent lines intersect at right angles, the point of intersection must lie on the directrix of parabola.
Another bug on a parabola A bug is moving along the parabolay = x2. At what point on the parabola are the x- and y-coordinateschanging at the same rate? (Source: Calculus, Tom M. Apostol,Vol. 1, John Wiley & Sons, New York, 1967.)
Suspension Bridge A suspension bridge with weightuniformly distributed along its length has twin towers thatextend 75 meters above the road surface and are 400 metersapart. The cables are parabolic in shape and are suspendedfrom the tops of the towers. The cables touch the roadsurface at the center of the bridge. Find the height of thecables at a point 100 meters from the center. (Assume thatthe road is level.)
The paraboloid z= 6-x-x2 -2y2 intersects the plane x=1 in a parabola,Find parametric equations for the tangent line to this parabola at the point (1,2,-4) . Use a computer to graph the paraboloid,the parabola,and the tangent line on the same screen.
tangent line on the same screen.
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Finding The Focus and Directrix of a Parabola - Conic Sections; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=KYgmOTLbuqE;License: Standard YouTube License, CC-BY