If (r,theta)(r,θ) is in polar form and (x,y)(x,y) in Cartesian form the relation between them is as follows:
x=rcosthetax=rcosθ, y=rsinthetay=rsinθ, r^2=x^2+y^2r2=x2+y2 and tantheta=y/xtanθ=yx
Or, costheta=x/rcosθ=xr, sintheta=y/rsinθ=yr, theta=tan^(-1)(y/x)θ=tan−1(yx) and cottheta=x/ycotθ=xy.
Hence, r(2-costheta)=2r(2−cosθ)=2 can be written as
2r-rcostheta=22r−rcosθ=2
2(x^2+y^2)^(1/2)-x=22(x2+y2)12−x=2 or
2(x^2+y^2)^(1/2)=2+x2(x2+y2)12=2+x or
4(x^2+y^2)=(2+x)^24(x2+y2)=(2+x)2 or
4x^2+4y^2=4+x^2+4x4x2+4y2=4+x2+4x or
3x^2+4y^2-4x-4=03x2+4y2−4x−4=0
As coefficients of x^2x2 and y^2y2 are both positive but not equal, this is an ellipse.
The above can be written as
3(x^2-4/3x+4/9)+4y^2-4-12/9-03(x2−43x+49)+4y2−4−129−0
or 3(x-2/3)^2+4(y-0)^2=48/9=16/33(x−23)2+4(y−0)2=489=163
or 9/16(x-2/3)^2+3/4(y-0)^2=1916(x−23)2+34(y−0)2=1
or (x-2/3)^2/(16/9)+(y-0)^2/(4/3)=1(x−23)2169+(y−0)243=1
Center of ellipse is (2/3,0)(23,0)
Major axis is 2xx4/3=8/32×43=83 and minor axis is 2xx2/sqrt3=4/sqrt32×2√3=4√3
graph{3x^2+4y^2-4x-4=0 [-3, 3, -1.5, 1.5]}
