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Chapter 10: Q. 12 (page 699)

In Problems11-18, find an equation of the conic described. Graph the equation by hand.

Hyperbola; center at(0,0); focus at(0,4); vertex at (0,-2)

Short Answer

Expert verified

The equation of the hyperbola isy216-x212=1

Step by step solution

01

Step 1. Given Information

Center at (0,0)(0,0)

Focus at (0,4)

Vertex at (0,-2)

02

Step 2. Calculation

Here, a=4

and -ae=-2

Therefore,e=12

03

Step 3. Calculation

Now, a2(1-e2)=42(1-14)

=12

04

Step 4. Final Answer

The equation of the hyperbola isy216-x212=1

05

Step 4. Graph

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Most popular questions from this chapter

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

2x2-3xy+4y2+2x+3y-5=0

Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand.

Focus at0,2and its vertex at0,0.

Find an equation for each ellipse. Graph the equation by hand.

Center at(1,2)vertex at(1,4) contains the point(1+3,3).

In Problems 67鈥�74, analyze each conic

x-324-y225=1

Rutherford鈥檚 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鈥檚 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.
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