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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

DXT
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1 Answers1

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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 $t\in [0,1]$.

Arnaud Mortier
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  • @Arnuad Mortier how can i calculate end point between line and sphere.please explain. – DXT Feb 07 '18 at 00:11
  • @DurgeshTiwari you know that an arbitrary point on the segment has coordinates $(12t+27(1-t),\ldots,\ldots)$. If you plug these coordinates into the equation of the sphere, then you will get the value of $t$ corresponding to the intersection point. – Arnaud Mortier Feb 07 '18 at 10:20
  • @DurgeshTiwari and in fact you don't need to compute distances if you understand that the parameter $t$ can give you the ratio. – Arnaud Mortier Feb 07 '18 at 10:22