Correct choice is (a) \([±\frac{2}{\sqrt{5}}, 0, ∓\frac{1}{\sqrt{5}}] \)
For explanation: A cone is a level surface say f = 0 of f (x, y, z) = 4(x^2+y^2) – z^2
\(∴ ∇ f = 8x\hat{i} + 8y\hat{j} – 2z\hat{k} \)
or \( ∇ f_{(1,0,2)} = 8\hat{i} – 4\hat{k} \)
\(\hat{n} = \frac{(8\hat{i} – 4\hat{k})}{\sqrt{(8)2+(-4)2}} \)
\( = \frac{(8\hat{i} – 4\hat{k})}{\sqrt{80}} \)
\( = \frac{(8\hat{i} – 4\hat{k})}{\sqrt{(16*5)}} = \frac{(2\hat{i} – \hat{k})}{\sqrt{5}} \, or \, \hat{n} = [\frac{2}{\sqrt{5}}, 0, \frac{-1}{\sqrt{5}}] \)
There are two possible unit vectors, if one is along direction then \(\hat{n} \) other will be along – \(\hat{n} \)
\(∴ \hat{n} = [±\frac{2}{\sqrt{5}}, 0, ∓\frac{1}{\sqrt{5}}]. \)