SOLUTION: What is the graph of the polar equation r = 2 sec (theta + 90degrees)

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Question 1023362: What is the graph of the polar equation r = 2 sec (theta + 90degrees)
Found 2 solutions by ikleyn, robertb:
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
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What is the graph of the polar equation r = 2 sec (theta + 90degrees)
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As the first step, let us write this equation in (x,y) rectangular coordinates. It is 

sqrt%28x%5E2+%2B+y%5E2%29 = 2%2Fcos%28theta%2B90%5Eo%29,   or

sqrt%28x%5E2+%2B+y%5E2%29 = -2%2Fsin%28theta%29,   ( since cos%28theta%2B90%5Eo%29 = -sin%28theta%29 ),   or

sqrt%28x%5E2+%2B+y%5E2%29 = -2%2Fy   ( since sin%28theta%29 = y ).

To rid off the square root, square both sides. You will get

x%5E2+%2B+y%5E2 = 4%2Fy%5E2.

If you want to express y via x, multiply both sides by y%5E2 and simplify. You will get

y%5E4+%2B+x%5E2y%5E2+-+4 = 0.

Consider it as a quadratic equation for y%5E2. Express y%5E2 as its root, using the quadratic formula 

y%5E2 = %28-x%5E2+%2B-+sqrt%28x%5E4+-+4%2A%28-4%29%29%29%2F2 = %28-x%5E2+%2B-+sqrt%28x%5E4+%2B+16%29%29%2F2.

Then  y = +/- sqrt%28sqrt%28x%5E4%2B16%29-x%5E2%29%2F2%29.


 
  

   




        Plots y = sqrt%28sqrt%28x%5E4%2B16%29-x%5E2%29%2F2%29

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
r+=+2sec%28theta+%2B+90%5Eo%29
<==> r+=+2%2Fcos%28theta+%2B+90%5Eo%29
<==> r+=+-2%2Fsin%28theta%29
<==> rsin%28theta%29+=+-2
<==> y = -2, by direct application of polar-cartesian conversion law.
==> the graph of the equation is a horizontal line with y-intercept at y = -2.