$\"\"$

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The parabolic cylinders equations are $\"\"$ and $\"\"$ and the point is $\"\"$ .

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Let us assume $\"\"$ .

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$\"\"$ and $\"\"$ .

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Therefore, the parametric equations are $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$ for the point $\"\"$ is when $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side.

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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The binomial vector is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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The binomial vector is $\"\"$ .

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The normal vector is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$ .

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The normal vector is $\"\"$ .

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$\"\"$

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The normal plane has a normal vector $\"\"$ and the point is $\"\"$ .

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The equation of the normal plane is $\"\"$

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$\"\"$

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The equation for normal plane is $\"\"$ .

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The oscillating plane has a binomial vector $\"\"$ and the point is $\"\"$ .

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The equation of the oscillating plane is $\"\"$

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$\"\"$

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Therefore, the equation of the osculating plane is $\"\"$ .

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$\"\"$

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The equation for normal plane is $\"\"$ and the equation of the osculating plane is $\"\"$ .