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Find equations for the tangent plane to the surface z = x2 + y2 at the point (2, -1, 5).
- a)Tangent plane ⇒ 4x - 2y - z = 5
- b)Tangent plane ⇒ 4x + 2y - z = 5
- c)Tangent plane ⇒ 4x - 2y + z = 5
- d)Tangent plane ⇒ -4x - 2y - z = 5
Correct answer is option 'A'. Can you explain this answer?
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Find equations for the tangent plane to the surface z = x2 + y2 at the...
Surface z = x2 + y2
⇒ surface S = x2 + y2 - z = 0
hence
hence
(1)the equation of a plane, passing through a point whose position vector is r0 and which is perpendicular to the normal N
(r - r0).N = 0
then the required equation is
⇒ 4(x - 2) - 2(y + 1) - 1 (z - 5) = 0⇒ 4 x - 8 - 2 y - 2 - z + 5 = 0
⇒ 4x - 2y - z = 5
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Find equations for the tangent plane to the surface z = x2 + y2 at the...
To find the equation of the tangent plane to the surface z = x^2 * y^2 at the point (2, -1, 5), we first need to find the partial derivatives of z with respect to x and y.
∂z/∂x = 2xy^2
∂z/∂y = 2x^2y
Next, we evaluate these partial derivatives at the point (2, -1, 5):
∂z/∂x = 2(2)(-1)^2 = 4
∂z/∂y = 2(2^2)(-1) = -8
The equation of the tangent plane can then be written as:
z - z0 = ∂z/∂x * (x - x0) + ∂z/∂y * (y - y0)
Where (x0, y0, z0) is the given point (2, -1, 5).
Plugging in the values:
z - 5 = 4(x - 2) - 8(y - (-1))
Simplifying:
z - 5 = 4x - 8 - 8y + 8
Rearranging the terms:
4x - 8y + z = 1
Therefore, the equation of the tangent plane is 4x - 8y + z = 1.
∂z/∂x = 2xy^2
∂z/∂y = 2x^2y
Next, we evaluate these partial derivatives at the point (2, -1, 5):
∂z/∂x = 2(2)(-1)^2 = 4
∂z/∂y = 2(2^2)(-1) = -8
The equation of the tangent plane can then be written as:
z - z0 = ∂z/∂x * (x - x0) + ∂z/∂y * (y - y0)
Where (x0, y0, z0) is the given point (2, -1, 5).
Plugging in the values:
z - 5 = 4(x - 2) - 8(y - (-1))
Simplifying:
z - 5 = 4x - 8 - 8y + 8
Rearranging the terms:
4x - 8y + z = 1
Therefore, the equation of the tangent plane is 4x - 8y + z = 1.
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Find equations for the tangent plane to the surface z = x2 + y2 at the point (2, -1, 5).a)Tangent plane ⇒ 4x - 2y - z = 5b)Tangent plane ⇒ 4x + 2y - z = 5c)Tangent plane ⇒ 4x - 2y + z = 5d)Tangent plane ⇒ -4x - 2y - z = 5Correct answer is option 'A'. Can you explain this answer?
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