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# Find the equation for the tangent plane at each point.?

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- ted sLv 77 months agoFavorite Answer
the gradient is < 2 x / 9 , y / 2 , 2 z >...

at (3 , 0 , 0 ) the tangent plane is : ( 2 / 3 ) ( x - 3 ) = 0 ;

at ( 0 , 2 , 0 ) the tangent plane is : ( y - 2 ) = 0 ;

at ( 0 , 0 , -1 ) the tangent plane is - 2 ( z + 1 ) = 0 ;

at ( 1 , 1 , - √23 / 6 ) the tangent plane is

( 2/9) ( x - 1) + ( 1 / 2 ) ( y - 1 ) - (√23 / 3 ) ( z + √23 / 6 ) = 0..

note that it is an ellipsoid , NOT an ellipse

- Φ² = Φ+1Lv 77 months ago
The tangent plane to the spheroid (x/a)² + (y/b)² + (z/c)² = 1 at the tangent point (x₀, y₀, z₀) is (x₀/a²)x + (y₀/b²)y + (z₀/c²)z = 1. ᵠ

Here for our ellipsoid this is (x₀/9)x + (y₀/4)y + (z₀)z = 1.

Kindly substitute in your chosen tangent points, and simplify.

Source(s): ᵠ: The same way the tangent line to a circle x²+y²=r² at the tangent point (x₀,y₀) is x₀x+y₀y=r², or the tangent line to an ellipse (x/a)²+(y/b)²=1 at the tangent point (x₀,y₀) is (x₀/a²)x+(y₀/b²)y=1, or the tangent plane to the sphere x²+y²+z²=r² at the tangent point (x₀,y₀,z₀) is x₀x+y₀y+z₀z=r². This is single power substitution for the tangent for specific shapes/solids.

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