If (r,θ) is in polar form and (x,y) in Cartesian form the relation between them is as follows:
x=rcosθ, y=rsinθ, r2=x2+y2 and tanθ=yx
Or, cosθ=xr, sinθ=yr, θ=tan−1(yx) and cotθ=xy.
Hence, r(2−cosθ)=2 can be written as
2r−rcosθ=2
2(x2+y2)12−x=2 or
2(x2+y2)12=2+x or
4(x2+y2)=(2+x)2 or
4x2+4y2=4+x2+4x or
3x2+4y2−4x−4=0
As coefficients of x2 and y2 are both positive but not equal, this is an ellipse.
The above can be written as
3(x2−43x+49)+4y2−4−129−0
or 3(x−23)2+4(y−0)2=489=163
or 916(x−23)2+34(y−0)2=1
or (x−23)2169+(y−0)243=1
Center of ellipse is (23,0)
Major axis is 2×43=83 and minor axis is 2×2√3=4√3
graph{3x^2+4y^2-4x-4=0 [-3, 3, -1.5, 1.5]}
