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The equation for the tangent plane is 12x - y + 4z - 56 = 0 and the equation for normal line is (x - 6)/12 = (32 - y) = (z - 2)/4
Finding the Equations for Tangent Plane and Normal Line:
The given function is,
f(x, y, z) = x² - y - z² = 0
∂f/ ∂x = 2x
∂f/ ∂y = -1
∂f/ ∂z = 2z
At given point (6, 32, 2),
∂f/ ∂x = 12
∂f/ ∂y = -1
∂f/ ∂z = 4
(a) The equation of tangent plane is given as follows,
(∂f/ ∂x)(x-x₁) + (∂f/ ∂y)(y-y₁) + (∂f/ ∂z)(z-z₁) = 0
12(x - 6) - 1(y - 32) + 4(z - 4) = 0
12x - 72 - y + 32 + 4z - 16 = 0
The required tangent plane is,
12x - y + 4z - 56 = 0
(b) The equation for normal line is given as,
(x-x₁) / (∂f/ ∂x) = (y-y₁) / (y-y₁) = (z-z₁) / (∂f/ ∂z)
(x - 6)/12 = (y - 32)/(-1) = (z - 2)/4
Thus, the required equation of normal line is,
(x - 6)/12 = (32 - y) = (z - 2)/4
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