$\"\"$

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The equation of the ellipse is $\"\"$ .

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The osculating circle has a radius at the origin is $\"\"$ .

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Find the curvature $\"\"$ .

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

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$\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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Again apply derivative on each side with respect to $\"\"$ .

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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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Find the curvature $\"\"$ . \ \

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

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$\"\"$

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$\"\"$ \ \

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Therefore the radius of osculating circle will be $\"\"$ . $\"\"$ \ \

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Find the osculating circle at $\"\"$ .

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$\"\"$

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The normal to $\"\"$ is $\"\"$ -axis.

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Centre of the osculating ellipse is $\"\"$ .

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Equating of osculating circle: $\"\"$

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$\"\"$ .

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Find the osculating circle at $\"\"$ .

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$\"\"$

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The normal to $\"\"$ is $\"\"$ -axis.

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Centre of the osculating ellipse is $\"\"$ .

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Equating of osculating circle: $\"\"$

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$\"\"$ .

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Graph:

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Graph the ellipse equation and osculating equations.

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$\"\"$ .

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$\"\"$

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The osculating equations are $\"\"$ and $\"\"$ .

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Graph the ellipse equation and osculating equations.

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$\"\"$ .