Final answer:
To find the equation of the tangent plane to the given surface, we need to find the partial derivatives and use them to construct the equation. Evaluating the partial derivatives at the given point, the equation of the tangent plane is 14x - 108y - 156z - 662 = 0.
Step-by-step explanation:
To find the equation of the tangent plane to the given surface, we need to find the partial derivatives and use them to construct the equation.
The given surface is x² - 3y² z² yz = 52. Taking the partial derivative with respect to x, we get 2x. Taking the partial derivative with respect to y, we get -6yz² - 3y²z. Taking the partial derivative with respect to z, we get -6y²z - 3y²z². Evaluating these partial derivatives at the point (7, 2, -5), we get 14, -108, and -156.
Using the point-normal form of the equation of a plane, the equation of the tangent plane is given by: 14(x - 7) - 108(y - 2) - 156(z + 5) = 0. Simplifying, we get 14x - 98 - 108y + 216 - 156z - 780 = 0. Combining like terms, the equation of the tangent plane is 14x - 108y - 156z - 662 = 0.