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=k
now putting the value of x, y and z in the above equation of the sphere we get
k2+k2+(k−1)2−k−k+k−1−1=03k2−3k−1=0
k=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)
distance between these points is (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
diameter of the sphere = s×radius=2×1.32=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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