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Chapter 8: Problem 29

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=3 \cos \theta, \quad y=3 \sin \theta $$

Short Answer

Expert verified
The points of horizontal tangency to the curve are at (0,3) and (0,-3) while the points of vertical tangency to the curve are at (3,0) and (-3,0).

Step by step solution

01

Find Derivatives

Compute the derivatives of x and y with respect to \( \theta \).\n\( dx/d\theta = -3 \sin \theta \)\n\( dy/d\theta = 3 \cos \theta \).
02

Find Values for Horizontal Tangency

For horizontal tangency, we need to find the values of \( \theta \) when \( dy/d\theta = 0 \).\nSolving \( dy/d\theta = 3 \cos \theta = 0 \) gives \( \theta = \pi/2 \), or \( 3\pi/2 \).
03

Find Points of Horizontal Tangency

Substitute these \( \theta \) values into the original equations to find the x and y coordinates of these points.\nFor \( \theta = \pi/2 \), \( x = 3 \cos(\pi/2) = 0 \), and \( y = 3 \sin(\pi/2) = 3 \), hence the point is (0,3).\nFor \( \theta = 3\pi/2 \), \( x = 3 \cos(3\pi/2) = 0 \) and \( y = 3 \sin(3\pi/2) = -3 \), the point is (0,-3).
04

Find Values for Vertical Tangency

For vertical tangency, we need to find the values of \( \theta \) when \( dx/d\theta = 0 \).\nSolving \( dx/d\theta = -3 \sin \theta = 0 \) gives \( \theta = 0 \), or \( \pi \).
05

Find Points of Vertical Tangency

Substitute these \( \theta \) values into the original equations to find the x and y coordinates of these points.\nFor \( \theta = 0 \), \( x = 3 \cos(0) = 3 \), and \( y = 3 \sin(0) = 0 \), hence the point is (3,0).\nFor \( \theta = \pi \), \( x = 3 \cos(\pi) = -3 \) and \( y = 3 \sin(\pi) = 0 \), the point is (-3,0).

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Most popular questions from this chapter

Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=e^{a \theta} & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r(2+\sin \theta)=4\)

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.
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