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# The graph of $r = \frac{4}{2 - \cos \theta}$ forms a closed curve. The area of the region inside the curve can be expressed in the form \$k

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he graph of $$r = \frac{4}{2 - \cos \theta}$$

forms a closed curve. The area of the region inside the curve can be expressed in the form $$k \pi$$. What is $$k^2$$?

Dec 4, 2018

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Rekt.....let's put this in Cartesian form....maybe it will give us a clue as to what the graph might look like

√(x^2+ y^2)  =                4

______________

2√(x^2 + y^2) -  x

_______________

√(x^2 + y^2)

√ (x^2 + y^2)  =   4 √(x^2 + y^2)

_______________

2√(x^2 + y^2) - x

1           =               4

______________

2√ (x^2 + y^2) - x

2 √(x^2 + y^2) - x =  4

√(x^2 + y^2) - x/2 = 2

√(x^2 + y^2) =   2 + x/2     square both sides

x^2 + y^2 =   4 + 2x + x^2/4        multiply through by 4

4x^2 + 4y^2 = 16 + 8x + x^2    simplify

3x^2 - 8x + 4y^2 =16          complete the square on x

3(x^2 - (8/3)x +  16/9   -16/9 ) + 4y^2 = 16

3(x - 4/3)^2 + 4 y^2  =  16 + 48/9

3(x - 4/3)^2  + 4y^2  =  64/3            multiply both sides by 3/64

(9/64) (x - 4/3)^2 + (3/16)y^2  =  1

(x - 4/3)^2             y^2

________   +      _____  =       1

64/9                  16/3

This is an ellipse centered at (4/3, 0)

a = √[64/ 9] =  8/3

b = √[16/3] =  4/√3

The area is given by

pi * a * b   =

pi * 8/3 * 4/√3    =

pi * 32√3 / 9

k =  32√3 / 9

k^2 =    1024 / 27   Dec 4, 2018