Question

Find an equation of the tangent plane to the given surface at the specified point. z = 2(x - 1)² + 5(y - 3)² - 2, (2, -2, 9)

Answer 1

The equation of the tangent plane to the surface
`\(z = 2(x - 1)^2 + 5(y - 3)^2 - 2\) at the point (2, -2, 9) is \(2x - 4y + z - 17 = 0\)` .

Step-by-step explanation:

To find the equation of the tangent plane to the surface at a specified point, we first need the surface equation and the point coordinates. The equation of the tangent plane at point
`\((x_0, y_0, z_0)\)` on a surface defined by
`\(F(x, y, z) = 0\)` 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
`\(F_x\), \(F_y\), and \(F_z\)` are partial derivatives of
`\(F\)` with respect to
`\(x\), \(y\)` , and
`\(z\)` respectively.

Given
`\(z = 2(x - 1)^2 + 5(y - 3)^2 - 2\),` we calculate partial derivatives:

`\(F_x = 4(x - 1)\) and \(F_y = 10(y - 3)\)` .

At point
`(2, -2, 9)` :

`\(F_x(2, -2, 9) = 4(2 - 1) = 4\)` and
`\(F_y(2, -2, 9) = 10(-2 - 3) = -50\)` .

Also,
`\(F(2, -2) = 2(2 - 1)^2 + 5(-2 - 3)^2 - 2 = 2 + 5(25) - 2 = 125\)` .

Substituting into the equation of the tangent plane:

`\(4(x - 2) - 50(y + 2) + (z - 9) = 0\)` .

Rearranging terms:
`\(4x - 8 - 50y - 100 + z - 9 = 0\)`
`\(4x - 8 - 50y - 100 + z - 9 = 0\)` .

Simplifying gives the equation of the tangent plane:
`\(4x - 50y + z - 117 = 0\)` .

Further simplification yields
`\(2x - 25y + z - 58.5 = 0\)` , which can be

multiplied by 2 to remove fractions:
`\(2x - 50y + 2z - 117 = 0\)` .

Thus, the equation of the tangent plane is
`\(2x - 4y + z - 17 = 0\)` .