An equation of the hyperbola with center at (0, 0), foci at (-c, 0) and (c, 0),
ID: 3109560 • Letter: A
Question
An equation of the hyperbola with center at (0, 0), foci at (-c, 0) and (c, 0), and vertices at (-a, 0) and (a, 0) is: x^2/a^2 - y^2/b^2 = 1, where b^2 = c^2 - a^2 The hyperbola x^2/a^2 - y^2/b^2 = 1 has the two oblique asymptotes: y = b/a x and y = -b/a x. In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. The figure shows a diagram from the scientist's paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola. (a) Find an equation of the asymptotes under this scenario. (b) If the vertex of the path of the alpha particles is 3 cm from the center of the hyperbola, find a model that describes the path of the particle.Explanation / Answer
Since the angle of the asymptotes or the slope is mentioned as 45 degrees
Therefore, the equation of asymptotes are y = x and y = -x
=> b = a
The equation of hyperbola of the particle would be : x2 - y2 = c2
Since, it is given that the equation of path passes through (3,0) we check it by plugging this value
=> 32 - 02 = c2
=> c = 3, -3
=> Equation of path of particle => x2 - y2 = 32
=> Equation of path of particle => x2 - y2 = 9