   Chapter 12.5, Problem 13E ### 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 + 2 3 x y − y 2 = 4 , ϕ = 30 ∘

To determine

The equation of the conic x2+23xyy2=4 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=30.

Explanation

Given:

The equation of the conic x2+23xyy2=4 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=30.

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.

Write the formula of the inverse trigonometry.

sin2θ=1cos2θ

cos(cos1x)=x

Calculation:

Substitute 30 for ϕ in equation (1).

x=Xcos(30)Ysin(30)=X(32)Y(12)=3XY2

Similarly, Substitute 30 for ϕ in equation (2).

y=Xsin(30)+Ycos(30)=Xsin(30)+Ycos(30)=X(12)+Y(32)=X+3Y2

Now,

Substitute 3XY2 for x and X+3Y2 for y in equation x2+23xyy2=4

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