Given an ellipsoid with the formula: $$\mathcal{I}: \ \ \frac{x^2}{25}+\frac{y^2}{16}+\frac{z^2}{9}=1$$ Find the equation of the chord that passes through $M(2,1,-1)$ and is divided equally with this point (i.e $M$ lies on the center of the segment of the line inside the ellipsoid)
As we know one point on the line we have $$\frac{x-2}{l}=\frac{y-1}{m}=\frac{z+1}{n}$$ where $(l,m,n)$ is the direction vector. As $M$ is right on the center we have two points $A(a_1,a_2,a_3)$ and $B(b_1,b_2,b_3)$ such that $\frac{A+B}{2}=M$ and $A,B \in \mathcal{I}$. Then $(l,m,n)=(b_1-a_1,b_2-a_2,b_3-a_3)$.
From $A,B \in \mathcal{I}$, we may obtain that $$\frac{4}{25}l+8m-\frac{2}{9}n=0$$ However still I can't complete the task, I can't see where the right track is... Any help is aprreciated.
Aside from what is likely a typo in your last equation, you’re on the right track. That last equation should be $$\frac4{25}l+\frac18m-\frac29n=0.$$
So far, you’ve determined that the chord must lie on the plane $\frac4{25}x+\frac18y-\frac29z={1201\over1800}$. From here, we can see that there isn’t a unique solution to the problem. The intersection of this plane turns out to be an ellipse centered at $M$, so every chord through $M$ that lies on this plane satisfies the conditions of the problem.