Question #102909
Show that x = y = z+1 is a secant line of the sphere x^2 +y^2 +z^2 −x−y+z−1 = 0.
Also find the intercept made by the sphere on the line.
1
Expert's answer
2020-02-20T08:30:46-0500

Let assume x=y=z+1=kx=y=z+1=k

now putting the value of x, y and z in the above equation of the sphere we get

k2+k2+(k1)2kk+k11=03k23k1=0k^2+k^2+(k-1)^2-k-k+k-1-1=0 \newline 3k^2-3k-1=0

k=1.26 &0.26=1.26 \ \& -0.26

so we get two points where the line cuts the sphere :

(1.26,1.26,0.26) and (0.26,0.26,1.26)(1.26,1.26,0.26)\ and \ (-0.26,-0.26,-1.26)

distance between these points is (1.26+0.26)2(1.26+0.26)2(1.260.26)2=2.63\sqrt{(1.26+0.26)^2(1.26+0.26)^2(-1.26-0.26)^2}=2.63


radius of the above sphere = (0.5)2+(0.5)2+(0.5)2+1=1.32\sqrt{(0.5)^2+(0.5)^2+(0.5)^2+1}=1.32

diameter of the sphere = s×radius=2×1.32=2.64s\times radius =2\times1.32=2.64


distance between the two points is less than the diameter of the sphere hence the given line is a secant the sphere and the intercept made by the line to the sphere is 2.63


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