# Consider the sphere S of radius 5 centred at (1, 0,2). Find the equation of the plane tangent to S at the point (5,-2,2). Hint: To get started, draw this situation in 2D with a circle.

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Expert Answers

sciencesolve | Certified Educator

You need to remeber that the directional vector of the line that links the center of the sphere with the tangency point is normal to the tangent plane to the sphere, hence, you need to evaluate the directional vector of the line passing through (1,0,2) and (5,-2,2)

such that:

`bar v = <5-1,-2-0,2-2> => bar v = <4,-2,0>`

You may write the equation of the tangent plane to the sphere, at the given tangency point, such that:

`4*(x - 5) - 2 (y + 2) + 0*(z- 2) = 0`

`4x - 20 - 2y - 4 = 0 => 4x - 2y - 24 = 0 => 2x - y - 12 = 0`

**Hence, evaluating the equation of the tangent plane to the sphere, at the point `(5,-2,2)` , yields **`2x - y - 12 = 0.`

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