Q.Two galaxies of masses 9 billion solar mass and 4 billion solar mass are 5 million light years apart. If the Sun has to cross the line joining them, without being attracted by either of them, through what point should it pass?

[!TLDR] The null point divides the 5-million-light-year separation in the ratio $\sqrt{M_1}:\sqrt{M_2}=3:2$, giving 3 million light years from the 9-billion-solar-mass gal …

At the point where the Sun feels zero NET gravitational pull from the two galaxies, the two attractions must be exactly equal and opposite: $\frac{GM_1m}{d_1^2}=\frac{GM_2m}{d_2^2}$ where $d_1$ is the distance from the 9-billion-solar-mass galaxy and $d_2$ from the 4-billion-solar-mass galaxy, with $d_1+d_2=5$ million light years. This gives $\frac{d_1}{d_2}=\sqrt{\frac{M_1}{M_2}}=\sqrt{\frac{9}{4}}=\frac{3}{2}$. Writing $d_1=3k$ and $d_2=2k$, the total $d_1+d_2=5k=5$ million light years gives $k=1$ million light years, so $d_1=3$ million light years and $d_2=2$ million light years. [!ANSWER] The Sun should pass through the …

Set the two gravitational forces (from each galaxy on the Sun) equal in magnitude, giving a distance ratio equal to the square root of th …

Using the mass ratio directly (9:4) instead of its SQUARE ROOT (3:2) to split the distance, since the …