. . Reflective Property of an Ellipse The lines from the foci to a point on an ellipse make equal angles with the tangent line at that point (Fig. .
). The light or sound or radio waves emitted from one focus hits any point P on the ellipse is received at the other focus (Fig. . ).
Example . A room m long is constructed to be a whispering gallery. The room has an elliptical ceiling, as shown in Fig. .
. If the maximum height of the ceiling is m , determine where the foci are located. Fig. .
Fig. . ◄ ◄ ◄ ◄ ◄ ◄ ◄ ◄ O S A ( , ) S' A' S O ◄ ◄ S S' ß α = ß P Kidney Stones Shock waves S' S ◄ ◄ ◄ ◄ ◄ ◄ Source - - The length a of the semi major axis of the elliptical ceiling is17 m . The height b of the semi minor axis is m .
Thus c = a = then c = For the elliptical ceiling the foci are located on either side about m from the centre, along its major axis. A non-invasive medical miracle In a lithotripter, a high-frequency sound wave is emitted from a source that is located at one of the foci of the ellipse. The patient is placed so that the kidney stone is located at the other focus of the ellipse. Example .
If the equation of the ellipse is x ) + ( x and y are measured in centimeters) where to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone? The equation of the ellipse is x ) + . The origin of the sound wave and the kidney stone of patient should be at the foci in order to crush the stones. a = and b c = a = = c .
Therefore the patient’s kidney stone should be placed .5cm from the centre of the ellipse.