   Chapter 12.5, Problem 9E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

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

# 9-14 ■Finding the Equation for a Rotated Conic Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. x 2 − 3 y 2 = 4 , ϕ = 60 ∘

To determine

The equation of the conic x23y2=4 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=60.

Explanation

Given:

The equation of the conic x23y2=4 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=60.

Approach:

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

x=XcosϕYsinϕ …… (1)

y=Xsinϕ+Ycosϕ …… (2)

Here, (x,y) is the coordinates of the point and ϕ is the angle at which coordinate axis are rotated.

Calculation:

Substitute 60 for ϕ in equation (1).

x=Xcos60Ysin60=X(12)Y(32)=X3Y2

Similarly, 60 for ϕ in equation (2).

y=Xsin60+Ycos60=X(32)+Y(12)=3X+Y2

Now,

Substitute X3Y2 for x and 3X+Y2 for y in equation x23y2=4

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