Ratio in which the sphere divides the line joining $A$ and $B$

Question

The ratio in which the sphere $x^2+y^2+z^2=504$ divides tbe line joining the points $A(12,-4,8)$ and $(27,-9,8)$ internally.

Try: let sphere divide line joining $A$ and $B$ in $\lambda:1$

Let $P$ be a point lie on line joining $AB$

So coordinate of $P$ is $$\bigg(\frac{27\lambda+12}{\lambda+1},\frac{-9\lambda-4}{\lambda+1},\frac{8\lambda+8}{\lambda+1}\bigg)$$

Could some help me to solve , thanks

Answer 1

Hint:

1. find the intersection point(s?) between sphere and line
2. compute the distance between that point and the endpoints of the segment
3. compute the ratio of distances

Hint2:

A point on a segment $[AB]$ has $x$-value $tx_A+(1-t)x_B$ for some $t\in [0,1]$.