Find the tangent plane to the ellipsoid?

Question

Find the tangent plane to the ellipsoid x^2+4y^2=169-9z^2 at the point P=(3, 2, 4).

Answer 1

Yes this is correct. We know this because any vector that lies on the tangent plane will be a tangent vector at the point (3,2,4). This is important because that gradient vector you found at (3,2,4) is perpendicular to the tangent vector which allows you to use it as your normal vector in the equation of the plane.

I would suggest going further with your answer by putting your equation in the form ax + by + cz = d unless you know your instructor will accept it the way it is.