# Elliptical orbits and Hohmann transfer

I had a rather theoretical problem with the elliptical velocity equation while glancing the formulas and proofs. While doing the proof there was a statement of energy conservation : $$K.E.+P.E.=M.E.$$ $$\frac{1}{2}mv^2-\frac{GMm}{r}=-\frac{GM}{2a}=\frac{-GM}{r+r_p}$${where v is the velocity of the object on the ellipse, a is semi major axis ,r is the distance of the object from the sun(or here the distance of the apogee) and $$r_p$$ is the distance of the perigee}

Now according to this if the velocity is increased $$r_p$$ should decrease therefore making the orbit smaller(since r is constant also shown in picture).

Mathematically: $$\frac{\frac{1}{2}mv^2}{GM}=\frac{1}{r}-\frac{1}{r+r_p},$$if v increases then: $$\frac{1}{r+r_p}$$ should also increase since $$\frac{1}{r}$$ is constant. This means that $$r_p$$ should decrease. Howewer in Hohmann's transfer orbit increasing the velocity should increase the eccentricity therefore increasing $$r_p$$.

What am i doing wrong here?

You have already gotten to the correct mathematical equation yet you are confused about the minus sign. $$\frac{\frac12mv^2}{GMm}=\frac1r-\frac1{r+r_P}$$ is correct. As $$v$$ increases, the RHS also increases. $$\frac1r$$ is a constant, so $$-\frac1{r+r_P}$$ has to increase, meaning $$\frac1{r+r_P}$$ decreases, $$r_P$$ increases. This agrees with the standard treatments saying that eccentricity increases.
Maybe you might like it better if you had rewritten it as $$r+r_P=\frac1{\frac1r-\frac{v^2}{2GM}}$$ so that, by dividing by a smaller number, you see that $$r_P$$ has to increase.