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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Strophoid The curve given by the parametric equations

x ( t ) = 1 t 2 1 + t 2 and y ( t ) = t ( 1 t 2 ) 1 + t 2

is called a strophoid.

(a) Find a rectangular equation of the strophoid.

(b) Find a polar equation of the strophoid.

(c) Sketch a graph of the strophoid.

(d) Find the equations of the two tangent lines at the origin.

(e) Find the points on the graph at which the tangent lines are horizontal.

(a)

To determine

To calculate: The rectangular equation of the Strophoid if

x(t)=1t21+t2y(t)=t(1t2)1+t2

Explanation

Given:

The given equations are:

x(t)=1t21+t2y(t)=t(1t2)1+t2

Calculation:

Square both the given equations

x2(t)=(1t21+t2)2x2(t)=(1t2)2(1+t2)2

And,

y2(t)=[t(1t2)1+t2]2y2(t)=t2(1t2)2(1+t2)2

To write t2 in rectangular form, simplify the function 1x1+x as shown below.

Put the value of x(t)=1t21+t2 in the function 1x(t)1+x(t)

(b)

To determine

To calculate: The polar equation of the Strophoid if

x(t)=1t21+t2y(t)=t(1t2)1+t2

(c)

To determine

To graph: The equation of strophoid if

x(t)=1t21+t2y(t)=t(1t2)1+t2

(d)

To determine

To calculate: The equation of two tangent lines at the origin if

x(t)=1t21+t2y(t)=t(1t2)1+t2

(e)

To determine

To calculate: The point at which the tangent lines are horizontal if

x(t)=1t21+t2y(t)=t(1t2)1+t2

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