$\"\"$

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The equation of the parabola 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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Apply derivative on each side with respect to $\"\"$ .

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

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$\"\"$

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

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$\"\"$

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Since the parabola opens upwards, 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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To get a radius of $\"\"$ , move to 2 units up and 2 units left from the initial point.

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Hence centre of the circle 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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$\"\"$ .