   Chapter 12.5, Problem 33E ### 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

# Identifying a Hyperbola Using Rotation of Axes(a) Use rotation of axes to show that the following equation represents a hyperbola. 7 x 2 + 48 e y − 7 y 2 − 200 x − 150 y + 600 = 0 (b) Find the X Y − and x y − coordinates of the center, vertices, and foci.(c) Find the equation of the asymptotes in X Y − and x y − coordinates.

To determine

(a)

To show:

The given equation represents a hyperbola using rotation of axes.

Explanation

Given:

The given equation is,

7x2+48xy7y2200x150y+600=0

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 plane are related as,

x=XcosϕYsinϕ(1)

y=Xsinϕ+Ycosϕ(2)

X=xcosϕ+ysinϕ

Y=xsinϕ+ycosϕ

Calculation:

Compare the given equation with the standard equation to get,

A=7,B=48,C=7

On rotation of axes, the acute angle ϕ satisfies,

cot2ϕ=ACB=724

Then,

cos2ϕ=725cosϕ=1+7252=45

sinϕ=35

Substitute 35 for sinϕ and 45 for cosϕ in equations (1) and (2)

To determine

(b)

To find:

The XY and xy coordinates of the center, vertices, and foci.

To determine

(c)

To find:

The equation of the asymptotes in XY and xy coordinates

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