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Chapter 10: Q. 49 (page 655)

Analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility.

2x2+3y2-8x+6y+5=0

Short Answer

Expert verified

The center of ellipse is at (2,-1), vertices are (2±3,-1)and the foci are (1,-1)and(3,-1)

The required graph is

Step by step solution

01

Step 1. Given Information 

The given equation is 2x2+3y2-8x+6y+5=0

02

Step 2. Calculation

Rewrite the given equation in the general form of equation of ellipse,

2x2-8x+3y2+6y=-52(x2-4x)+3(y2-2y)=-52(x2-4x+4)+3(y2+2y+1)=-5+8+32(x-2)2+3(y+1)2=62(x-2)26+3(y+1)26=66(x-2)23+(y-(-1))22=1

The major axis is parallel to x-axis.

Now, on comparing the obtained equation with the general form of equation, we get,

(h,k)=(2,-1)a=3,b=2c2=a2-b2c2=3-2c2=1c=1

The vertices of major axis are at(h±a,k)=(2-3,-1)and(2+3,-1)

And the foci are at point(h±c,k)=(1,-1)and(3,-1)

03

Step 3. Graph

On plotting all the points on the graph, we get the required graph as follows,

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Most popular questions from this chapter

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

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

In Problem, identify the graph of each equation without completing the squares.

2x2+y2-8x+4y+2=0

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

Center at (1,2): focus at(1,4): contains the point(2,2)

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.

In Problems 67–74, analyze each conic

x-324-y225=1
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