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**Finding the Equation of the Tangent Plane to a Surface** Given the surface equation: \[ x^2 + 8y^2 + z^2 = 468 \] Find the equation of the tangent plane at the point \((-6, 6, 12)\). **Possible Equations for the Tangent Plane:** A. \( 12(x - 6) - 96(y + 6) + 12(z - 12) = 0 \) B. \( 12(x - 6) - 96(y + 6) + 24(z - 12) = 0 \) C. \( 12(x - 6) - 96(y + 6) - 24(z - 12) = 0 \) D. \( 12(x - 6) - 48(y + 6) + 24(z - 12) = 0 \) E. \( 12(x + 6) + 96(y + 6) + 24(z - 12) = 0 \) **Note:** When finding the equation of the tangent plane to a given surface at a specific point, the normal vector to the surface must be determined at that point. This normal vector is essential for writing the plane’s equation, as it gives the coefficients of \(x\), \(y\), and \(z\) in the plane's equation. The options present different potential forms for this tangent plane equation.

**Finding the Equation of the Tangent Plane to a Surface** Given the surface equation: \[ x^2 + 8y^2 + z^2 = 468 \] Find the equation of the tangent plane at the point \((-6, 6, 12)\). **Possible Equations for the Tangent Plane:** A. \( 12(x - 6) - 96(y + 6) + 12(z - 12) = 0 \) B. \( 12(x - 6) - 96(y + 6) + 24(z - 12) = 0 \) C. \( 12(x - 6) - 96(y + 6) - 24(z - 12) = 0 \) D. \( 12(x - 6) - 48(y + 6) + 24(z - 12) = 0 \) E. \( 12(x + 6) + 96(y + 6) + 24(z - 12) = 0 \) **Note:** When finding the equation of the tangent plane to a given surface at a specific point, the normal vector to the surface must be determined at that point. This normal vector is essential for writing the plane’s equation, as it gives the coefficients of \(x\), \(y\), and \(z\) in the plane's equation. The options present different potential forms for this tangent plane equation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Finding the Equation of the Tangent Plane to a Surface** Given the surface equation: \[ x^2 + 8y^2 + z^2 = 468 \] Find the equation of the tangent plane at the point \((-6, 6, 12)\). **Possible Equations for the Tangent Plane:** A. \( 12(x - 6) - 96(y + 6) + 12(z - 12) = 0 \) B. \( 12(x - 6) - 96(y + 6) + 24(z - 12) = 0 \) C. \( 12(x - 6) - 96(y + 6) - 24(z - 12) = 0 \) D. \( 12(x - 6) - 48(y + 6) + 24(z - 12) = 0 \) E. \( 12(x + 6) + 96(y + 6) + 24(z - 12) = 0 \) **Note:** When finding the equation of the tangent plane to a given surface at a specific point, the normal vector to the surface must be determined at that point. This normal vector is essential for writing the plane’s equation, as it gives the coefficients of \(x\), \(y\), and \(z\) in the plane's equation. The options present different potential forms for this tangent plane equation.
Transcribed Image Text:**Finding the Equation of the Tangent Plane to a Surface** Given the surface equation: \[ x^2 + 8y^2 + z^2 = 468 \] Find the equation of the tangent plane at the point \((-6, 6, 12)\). **Possible Equations for the Tangent Plane:** A. \( 12(x - 6) - 96(y + 6) + 12(z - 12) = 0 \) B. \( 12(x - 6) - 96(y + 6) + 24(z - 12) = 0 \) C. \( 12(x - 6) - 96(y + 6) - 24(z - 12) = 0 \) D. \( 12(x - 6) - 48(y + 6) + 24(z - 12) = 0 \) E. \( 12(x + 6) + 96(y + 6) + 24(z - 12) = 0 \) **Note:** When finding the equation of the tangent plane to a given surface at a specific point, the normal vector to the surface must be determined at that point. This normal vector is essential for writing the plane’s equation, as it gives the coefficients of \(x\), \(y\), and \(z\) in the plane's equation. The options present different potential forms for this tangent plane equation.
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