. . Reflective Property of a Hyperbola The lines from the foci to a point on a hyperbola make equal angles with the tangent line at that point (Fig. .
). The light or sound or radio waves directed from one focus is received at the other focus as in the case ellipse (Fig. . ) used in spotting location of ships sailing in deep sea.
Example . Two coast guard stations are located km apart at points A ( , ) and B ( , . A distress signal from a ship at P is received at slightly different times by two stations. It is determined that the ship is km farther from station A than it is from station B .
Determine the equation of hyperbola that passes through the location of the ship. Fig. . Fig.
. Fig. . Elliptical ceiling of a whispering gallery m m S' S ◄ ◄ ◄ ◄ ◄ ◄ ◄ ◄ – + Ultrasound emitter Elliptic reflector Kidney stone Kidney S' (– c , ) S (– c , ) P ( x,y ) ß - - Two Dimensional Analytical Geometry - II Since the centre is located at ( , , midway between the two foci, which are the coast guard stations, the equation is y ) − ) = .
... ( ) To determine the values of a and b , select two points known to be on the hyperbola and substitute each point in the above equation. The point( , lies on the hyperbola, since it is km further from Station A than from station B . ) − 10000 .
There is also a point ( , x on the hyperbola such that600 360000 + x = x 40000 x = Substituting in ( ), we have 10000 ) − = 640000 = b = 80000 Thus the required equation of the hyperbola is y ) − 10000 80000 The