construct those wich are not huberbolas of f(x)= (mx+2)/((2-m)x+2m)

Question

M is a real parameter and i should find the curves of this function which are not hyperbolas

Answer 1

f(x)=(mx+2)/((2-m)x+2m).

When the denominator is a multiple of the numerator:

(2-m)x+2m=p(mx+2),

2x-mx+2m=pmx+2p,

2-m=pm, m=p (equating coefficients),

2-m=m², m²+m-2=0=(m+2)(m-1), so m=-2 or 1 gives us a linear function (horizontal line). Hence:

When m=1, f(x)=(x+2)/(x+2)=1 which is a linear function (horizontal line).

When m=-2, f(x)=(-2x+2)/(4x-4)=-½ which is a linear function (horizontal line).

When m=2 the denominator becomes a constant so f(x) is linear: f(x)=(2x+2)/4=½(x+2).

Other values of m produce hyperbolas.