An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

A) $8 x - 4 y + 12 z = 52$
B) $8 x - 4 y - 12 z = 52$
C) $8 x + 4 y + 12 z = 52$
D) $- 8 x - 4 y + 12 z = 52$
E) $- 8 x + 4 y + 12 z = 52$

Question

An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

A) $8 x - 4 y + 12 z = 52$
B) $8 x - 4 y - 12 z = 52$
C) $8 x + 4 y + 12 z = 52$
D) $- 8 x - 4 y + 12 z = 52$
E) $- 8 x + 4 y + 12 z = 52$

Quizplus · Accepted Answer

The equation of the tangent plane to a surface $F(x, y, z) = 0$ at a point $(x_0, y_0, z_0)$ is given by $F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0$, where $F_x, F_y, F_z$ are the partial derivatives of $F$ with respect to $x, y, z$, respectively. For the given surface $4x^2 + y^2 + 2z^2 = 26$, the partial derivatives at the point (1, -2, 3) are $F_x = 8x = 8$, $F_y = 2y = -4$, and $F_z = 4z = 12$. Substituting these values and the point (1, -2, 3) into the equation of the tangent plane gives $8(x - 1) - 4(y + 2) + 12(z - 3) = 0$, which simplifies to $8x - 4y + 12z = 52$.

An Equation of the Tangent Plane to the Surface 4x2+y2+2z2=264 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 264x2+y2+2z2=26

An Equation of the Tangent Plane to the Surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$