To see it write the equation equivalently:

x(ax+b) - y(cx+d) = x(x+(b/a)) - (c/a)y(x+d/c) = 0.

The equation represents a particular conic member of the family of conics generated by the two singular conics:

(i) c

(ii) c

The two degenerate conics c

g(x,y) = c

This shows that the graphs are conics and relates the value of (c/a) with the particular member passing through a definite point, say Q(1,k). Calculating g(1,k)=0 we find

a/c = k*(1+d/c)/(1+b/a).

Obviously cx+d=0 is an asymptote and this shows that the conic is a hyperbola.

The other asymptote assumed to be in the form y=Ax+B, results by finding {A,B} through limits. A is found by taking the limit f(x)/x and then B is found by taking the limit f(x)-Ax. Thus, A=a/c and B=(bc-ad)/c

The above formulas allow the determination of the asymptote y=Ax+B of the hyperbola, given the values of (-d/c), (-b/a) and k. In the formulas the symbols {dc, ba, ac} represent respectively the values {-d/c, -b/a, a/c}.

Hyperbola.html

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