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Chapter 10: Q. 37 (page 644)

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

x2=4y

Short Answer

Expert verified

The vertex is 0,0, focus is 0,1and directrix is y=-1.

The graph for the equation is shown below.

Step by step solution

01

Step 1. Given information .

Consider the given equationx2=4y.

02

Step 2. Analyze the equation

The standard form of parabola is x2=4ay.

Compare the equation with standard form,

x2=4ayx2=4y4a=4a=1

Further simplify.

Since a=1the vertex of parabola is 0,0,focus is role="math" localid="1646807678978" 0,a=0,1.

The directrix is y=-athat meansy=-1

03

Step 3. Plot the graph.

The graph of the parabola equation, x2=4yusing graphic utility is shown below.

Here, V is the vertex , F is focus and line y=-1represents the directrix of the parabola.

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Most popular questions from this chapter

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.

5x2+6xy+5y2-8=0

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 10cm from the center of the hyperbola, find a model that describes the path of the particle.

Prove that the hyperbola

y2a2-x2b2=1

has the two oblique asymptotesy=abxandy=-abx

In Problems 43–52, identify the graph of each equation without applying a rotation of axes.

4x2+12xy+9y2-x-y=0

True or False

The equation 3x2+Bxy+12y2=10defines a parabola ifB=-12.
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