I need help finding the equation of a Hyperbola?

Question

A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve.

Answer 1

dA = 3dB

dA = sqrt((x-(-1))^2+(y - 1)^2) =3dB

dB=sqrt((x-2)^2+(y-(-1))^2)

Square both equations

dB^2 = (x-2)^2+(y+1)^2

9dB^2 = (x+1)^2+(y-1)^2 = 9(x-2)^2 +9(y+1)^2

x^2+2x+1 + y^2-2y+1 = 9(x^2-4x+4 + y^2+2y+1)

x^2(1-9)+x(2+36)+2 + y^2(1-9)+y(-2-18)-36-9=0

-8x^2 + 38x - 8y^2 -20y -43 = 0

2x(-4x+19) - 4y(2y+5) = 43

Answer 2

why do you need to know not like you will ever have a practical use for it