1

I want to prove that the following are equivalent.

(a). The curve is part of a straight line.

(b). All its tangent lines are parallel.

(c). All its tangent lines pass through a fixed point $c$.

I was able to do (a) implies (b), but I'm having trouble coming up with a proof for (b) implies (c), and (c) implies (a). Could anyone help? Thanks!

gws
  • 639

1 Answers1

1

Let $(x(t),y(t))$ be a parametrization of the regular curve.

(b) to (c) :Suppose $c=(a,b)$ is the vector that the tangent lines are parallel to, then $$adx=bdy$$ which gives $y=\frac{b}{a}x+p$ which is a part of straight line.

(c) to (a):without losing generality suppose that all tangents come from the $(0,0)$ point then you have $(x,y)//(\frac{dx}{dt},\frac{dy}{dt})$ or $$xdy-ydx=0$$ or $\frac{xdy-ydx}{dx^2}=0$ or $(\frac{y}{x})'=0$ or $y=cx$ which is again part of straight line.

dmtri
  • 3,270