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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the parabolas r = c/(1 + cos θ) and r = d/(1 cos θ) intersect at right angles.

To determine

To find: Show the parabolas equation intersect at right angle.

Explanation

Given:

The given polar equation is r=c1+cosθ and r=d1cosθ

Calculation:

Consider the polar equation of r=c1+cosθ (1)

Differentiate the equation (1) with respect to θ

r=c[(1+cosθ)]1drdθ=c[(1)(1+cosθ)2(sinθ)]=csinθ(1+cosθ)2

Differentiate the above equation with respect to ‘x’.

dydx=drdθsinθ+rcosθdrdθcosθrsinθ=(csinθ(1+cosθ)2sinθ)+(c1+cosθcosθ)csinθ(1+cosθ)2cosθc1+cosθsinθ.(1+cosθ)2(1+cosθ)2

=csin2θ+ccosθ(1+cosθ)ccosθsinθcsinθ(1+cosθ)=ccosθ+c(sin2θ+cos2θ)ccosθsinθcsinθccosθsinθ=ccosθ+c(1)csinθ=c(cosθ+1)c(sinθ)

=

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