How do you convert #r=1/2costheta# into cartesian form?

1 Answer
Eddie
Jun 21, 2016

#(x - 1/4)^2 + y^2 = (1/4)^2 #

Explanation:

the basic ideas are #x = r cos theta, y = r sin theta# so #x^2 + y^2 = r^2 cos^2 theta + r^2 sin^2 theta = r^2 #

so using #r = sqrt{x^2 + y^2}# and #cos theta = x/r = x/sqrt{x^2 + y^2}# you can plug and play

So #sqrt{x^2 + y^2}= 1/2 x/sqrt{x^2 + y^2}#
#x^2 + y^2 = x/2#
#x^2 - x/2 + y^2 = 0#
completing the square in x
#(x - 1/4)^2 - 1/16 + y^2 = 0#
#(x - 1/4)^2 + y^2 = (1/4)^2 #

that's a circle centred on (1/4, 0) radius 1/4.

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