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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

Graphing an Equation Using Rotation of Axes. Show that the graph of the equation

x + y = 1

is part of a parabola by rotating the axes though an angle of 45 . [Hint: First convert the equation to one that does not involve radicals.]

To determine

To show:

The equation represents a parabola by rotating the axes.

Explanation

Given:

The given equation is,

x+y=1

Rotate the axes by an angle of 45.

Approach:

The standard equation of a conic is,

Ax2+Bxy+Cy2+Dx+Ey+F=0.

Suppose the x and y axes in a coordinate plane are rotated through the acute angle ϕ to produce the X and Y axes. Then, the coordinates (x,y) and (X,Y) of a point in the XY and xy planes are related as,

x=XcosϕYsinϕ(1)

y=Xsinϕ+Ycosϕ(2)

X=(xcosϕ+ysinϕ)

Y=(xsinϕ+ycosϕ)

Calculation:

First, remove the radicals from the given equations,

(x+y)2=12x+y+2xy=1(2xy)2=(1(x+y))24xy=1+(x+y)22(x+y)

1+x2+y2+2xy2x2y=4xy4xy1x2y22xy+2x+2y=0x22xy+y22x2y+1=0 (3)

Equation (3) represents the given equation after radicals are removed.

The axes have to be rotated by an angle of 45.

Substitute 45 for ϕ in equations (1) and (2) to get,

x=Xcos(45)Ysin(45)=X(12)Y(12)=XY2(4)

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