Question 11.3.5: Lithotripsy An elliptical water tank has a major axis of len...
Lithotripsy
An elliptical water tank has a major axis of length 6 feet and a minor axis of length 4 feet. The source of high-energy shock waves from a lithotripter is placed at one focus of the tank. To smash the kidney stone of a patient, how far should the stone be positioned from the source?
Because the length of the major axis of the ellipse is 6 feet, we have 2a = 6; so a = 3. Similarly, the minor axis of 4 feet gives 2b = 4, or b = 2. To find c, we use the equation c^{2}=a^{2}-b^{2}. We have
c^{2}=(3)^{2}-(2)^{2}=5 . \text { Therefore, } c=\pm \sqrt{5}.
If we position the center of the ellipse at (0, 0) and the major axis along the x-axis, then the foci of the ellipse are (-\sqrt{5}, 0) \text { and }(\sqrt{5}, 0) . The distance between these foci is 2 \sqrt{5} \approx 4.472 \text { feet } . The kidney stone should be positioned 4.472 feet from the source of the shock waves.