Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.
The vertex is ,the focus is and directrix is .
The graph of the equation shown below.
Consider the given equation .
The standard form of parabola is
Compare the equation with standard form
Further simplify .
Since the vertex is ,the focus is and the directrix is localid="1646988458852"
The graph of the equation using graphic utility is shown below .
Here V is the vertex , F is the focus and line represents the directrix of the parabola .
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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