Question

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.

Answer 1

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

$k^2+k^2+(k-1)^2-k-k+k-1-1=0 \newline 3k^2-3k-1=0$

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 $\sqrt{(1.26+0.26)^2(1.26+0.26)^2(-1.26-0.26)^2}=2.63$

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

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

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