Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand.
Vertex at , axis of symmetry is the x-axis; containing the point .
The equation of a parabola is . The points are and .
The graph of a parabola is :
The given vertex is at the point and the axis of symmetry is the x-axis and containing the point .
The vertex is at the origin, the axis of symmetry is the x-axis and the graph contains a point in the first quadrant. The general form of the equation is
Because the point is on the parabola, the coordinates must satisfy the equation of the parabola. Substitute the values, we get
The equation will be .
The focus is at the point . The two points that determines the latus rectum by letting . Then,
The points are and .
The graph of a parabola is
Rutherford’s Experiment 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. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherford’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 cm from the center of the hyperbola, find a model that describes the path of the particle.
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