Relation of speeds of Earth at extremities of orbit through force

Question

I came across this article in my book where they had related the velocities of earth at the aphelion and at the perihelion . Their approach was

$1)$ conservation of angular momentum at the desired points

$2)$ conservation of energy at the desired points.

This method is completely fine but I tried to think about it on my own and tried to replace the second step with equating force separately at the desired points. Now if we take the aphelion to be of distance $r_1$ where the speed of earth is $v_1$ and perihelion to be of distance $r2$ where the speed of earth is $v_2$ , through $1)$ we get $$r_1v_1=r_2v_2$$

And through second we get

$$\frac{G(m_s)(m_e)}{(r_1)^2} = \frac{(m_e)(v_1)^2}{r_1}$$

and

$$\frac{G(m_s)(m_e)}{(r_2)^2} = \frac{(m_e)(v_2)^2}{r_2}$$

Which gives us $$G(m_s) = (r_1)(v_1)^2=(r_2)(v_2)^2$$

Using the first result in the second result we get $v_1=v_2$. Which is obviously very very wrong untill and unless the orbit would have been circular. Now my question is , where is the flaw because I am unable to identify the step/part where I am doing the mistake.

Answer 1

To summarise the discussion in the comments , the formula $$F=\frac{GMm}{R^2}= \frac{mv^2}{R}$$ is only valid for circular orbits and is not valid for elliptical orbits , hence we are not able to relate the velocities of any two given points on the orbit via this method.