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Question. Suppose that all the tangent lines of a regular plane curve pass through some fixed point. Prove that the curve is part of a straight line. Prove the same result if all the normal lines are parallel.

I am working on differential geometry from the book by Pressley and I have a doubt in the solution of the above question whose (brief) solution is given by:

Solution: We can assume that the curve $\gamma$ is unit-speed and that the tangent lines all pass through the origin (by applying a translation to $\gamma$). Then, there is a scalar $\lambda(t)$ such that $\gamma'(t) = \lambda(t)\gamma(t)$ for all $t$. Then, $\gamma '' = \lambda'\gamma + \lambda \gamma' = (\lambda' + \lambda^2)\gamma$.

Can anyone please explain me how does this line follow : " Then, there is a scalar $\lambda(t)$ such that $\gamma'(t) = \lambda(t)\gamma(t)$ for all $t$."

Thanks in advance.