# The sphere x^2+y^2+z^2−2x+6y+14z+3=0 meets the line joining A(2,−1,4),B(5,5,5) in the points C and D. Prove that AC/CB=−AD/DB=1/2.

The sphere ${x}^{2}+{y}^{2}+{z}^{2}-2x+6y+14z+3=0$ meets the line joining $A\left(2,-1,4\right),B\left(5,5,5\right)$ in the points $C$ and $D$. Prove that $AC:CB=-AD:DB=1:2$
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Sandra Randall
Hint : The equation of the line through $A$ and $B$ is $\left(x,y,z\right)=\left(2+3t,-1+6t,4+t\right)$. Substitute this into the equation for the circle, solve the quadratic to find the points $C$ and $D$ ... etc ...
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Livia Cardenas
The line through $A\left(2,-1,4\right),B\left(5,5,5\right)$ has a direction given by $\stackrel{\to }{u}=B-A=\left\{3,6,1\right\}$
so the line $AB$ has equation $A+s\stackrel{\to }{v}=\left\{2+3s,-1+6s,4+s\right\}$
plug in the equation of the sphere
$\left(s+4{\right)}^{2}+\left(3s+2{\right)}^{2}+\left(6s-1{\right)}^{2}-2\left(3s+2\right)+6\left(6s-1\right)+14\left(s+4\right)+3=0$
$2\left(23{s}^{2}+26s+35\right)=0$
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