Question

convert 9(x-1)^2 + y^2=9 to polar form

Answer 1

y=rsinθ, x=rcosθ, tanθ=y/x, x²+y²=r².

9(x-1)²+y²=9 is an ellipse, centre (1,0).

9x²-18x+9+y²=9, 9x²-18x+y²=0, 9x²-18x+r²-x²=0, 8x²-18x+r²=0, 8r²cos²θ-18rcosθ+r²=0, 8rcos²θ-18cosθ+r=0, r=18cosθ/(8cos²θ+1). This can be written: r=18cosθ/(4cos(2θ)+5), by substituting cos²θ=½(cos(2θ)+1).