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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

Identifying a Parabola Using Rotation of Axes

(a) Use rotation of axes to show that the following equation represents a parabola.

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 vertex, and focus.

(c) Find the equation of the directrix in X Y and x y coordinates.

To determine

(a)

To show:

The given equation represents a parabola using rotation of axes.

Explanation

Given:

The given equation is,

22(x+y)2=7x+9y

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=22,B=42,C=22

On rotation of axes, the acute angle ϕ satisfies,

cot2ϕ=ACB=0

Then,

2ϕ=90ϕ=45cosϕ=12

sinϕ=12

Substitute 12 for sinϕ and 12 for cosϕ in equations (1) and (2)

To determine

(b)

To find:

The XY and xy coordinates of the vertex, and focus.

To determine

(c)

To find:

The equation of the directrix in XY and xy coordinates

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