Final answer:
The value of B in the tangent plane to the hyperboloid at the given point is -1/2√(21), and the value of D is -1/2√(7).
Step-by-step explanation:
To find the values B and D for the tangent plane equation x+B y+C z+D=0 at a given point on the hyperboloid 6 x²+3 y²-4 z²=51, we first need to compute the gradient (normal vector) of the hyperboloid at that point. The gradient vector can be found by taking the partial derivatives of the hyperboloid equation with respect to x, y, and z. The partial derivatives are 12x, 6y, and -8z, respectively. At the point (√(7),-√(3), 0), the gradient vector is (12√(7), -6√(3), 0). Since the gradient vector is normal to the tangent plane, and C is its z-component (which is 0 here), we can deduce that B corresponds to the y-component of the gradient normalized by the x-component. Therefore,
B = -6√(3) / 12√(7) = -√(3) / 2√(7),
which simplifies to
B = -1 / 2√(21).
To find D, we plug the point (√(7),-√(3), 0) into the equation x+By+Cz+D=0. Thus, we get
√(7) - (1 / 2√(21))(-√(3)) + 0 + D = 0
and solving for D yields
D = -√(7) + (1/2)(1/√(21))√(3),
which simplifies to
D = -√(7) + 1/2√(7) or D = -1/2√(7).