The largest and the shortest distances of the earth from the sun are r₁ and r₂. It’s distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun is

$\frac{2 r_1 r_2}{r_1+r_2}$

The equation of a general conic is $\frac{1}{r}=\frac{1}{\ell}(1+e \cos \theta)$, where e is eccentricity. For ellipse, turning points are at $\theta$ = 0, and $\theta$ = 180 giving $r_{\min }=r_2$, and $r_{\max}=r_1$ respectively.

∴  $\frac{1}{r_2}=\frac{1}{\ell}(1+e)$  and  $\frac{1}{r_1}=\frac{1}{\ell}(1-e)$

Add  $\frac{1}{r_2}+\frac{1}{r_1}=\frac{2}{\ell} ; 1=\frac{2 r_1 r_2}{r_1+r_2}$