## Question

###### 4. (20 points) Consider the polar curve r = 4cos 6 and r=1+2cos e. 1+2 caso...

4. (20 points) Consider the polar curve r = 4cos 6 and r=1+2cos e. 1+2 caso (a) Find ALL intersection points with 0 So< 2x and express them in polar coordi- nates. Show your works (b) Find the area inside the smaller loop of the curve r = 1+2 cose. Show your work. (c) Find the length of r = 4 cose from A to B as 8 increases, where A is the inter- section of the two curves in quadrant IV, and B is the intersection of the curve r= 4 cos with the positive r-axis. Show your work. 2 (d) Find the points where the tangent line to the curve r = 4 cos is horizontal and express the points in (r.) coordinates with 0 <o<27. Show your work.

## Answers

solutions © Given 8= 4COSO 8-1+2 COSO 81 All intersection with points ०८८27 1 +2 OSO 8= 4 COS Take equate them, 4 COSO z 4+2 oso 2 COSA - COSO TT -TT 올 플 Therefore, intersection point (8,0) ere I ()$(4, 5) smaller loop (b) Aree inside re 1+2coso Here, curve +2COS intersect x-Quis O=0 ? 8=2 1+2COSO 1 +2COSO - 3 Therefore, Are Q: N coso - 블 0-1 120,2409 Ź (1+2coso) do 4/3 옻 [sin ax + 4 sinx + 3x] 27/3 1.087 2 TT → area 05435to B (c length of 8 = 4 coso from a Here Are length of curve (8:4COSO) means i $ from -TT 3 O do do -1/2 como sea sino) 7 (41056)+ so O -IT/3 s. sino + coso do -T/ as in do 400-(-5) -T13 Arc length (d) points where tangent to 8= 4 caso id horizontal points in (0, 0) 4 TT 3 1 Curve Curve 7 4 COSO. Take, ç horizonta will be draw when 8=2 tangent 2. in carre 84coso Here, put 25 4 COSO 31 3 coso : { Therefore, I points are هلو (3, 5),(2,-5)

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