To prove the sum of all the intercepts = k^2
xi + yj +zk = k^2
The gradient of the function f(x,y,z) = s
f(x,y,z = 1/2sxi + 1/2 syj =1/2 szk
And the tangent plane at the point (xo,yo,zo) is
sx/x + sy/y + sz/z
= sxo/xo + syo/yo + szo/zo
=> sxo/xo + syo/yo + szo/zo = k ........(1)
To find the x intercept , we set y =z=0 and obtain x = ksxo/sx. similarly, y = ksyo/yo and z= kszo/zo. their sum is
ksxo/xo + ksyo/yo + kszo/zo
=k( sx0/xo + syo/yo + szo/zo)
=k.k = k^2 ...from eqn 1
To learn more about intercepts here
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