From the given, (x-1)^2+y^2=5(x−1)2+y2=5
Use x=r cos thetax=rcosθ and y=r sin thetay=rsinθ
Substitute these in the given variables x and y
(x-1)^2+y^2=5(x−1)2+y2=5
(r cos theta-1)^2+(r sin theta)^2=5(rcosθ−1)2+(rsinθ)2=5
r^2 cos^2 theta-2 r cos theta+1+r^2 sin^2 theta=5r2cos2θ−2rcosθ+1+r2sin2θ=5
factor out r^2r2
r^2(cos^2 theta+sin^2 theta)-2 r cos theta+1-5=0r2(cos2θ+sin2θ)−2rcosθ+1−5=0
simplify
r^2(1)-2 r cos theta-4=0r2(1)−2rcosθ−4=0
r^2-2 r cos theta-4=0r2−2rcosθ−4=0
Use now Quadratic Equation to solve for rr in terms of the other variable
r=(-b+-sqrt(b^2-4*a*c))/(2*a)r=−b±√b2−4⋅a⋅c2⋅a
using a=1a=1, and b=-2 cos thetab=−2cosθ,and c=-4c=−4
r=(-(-2 cos theta)+-sqrt((-2 cos theta)^2-4*1*(-4)))/(2*1)r=−(−2cosθ)±√(−2cosθ)2−4⋅1⋅(−4)2⋅1
r=(2cos theta+-sqrt((4 cos^2 theta+16)))/2r=2cosθ±√(4cos2θ+16)2
which simplifies to
r=cos theta+-sqrt(cos^2 theta+4)r=cosθ±√cos2θ+4
have a nice day!
