$\"\"$

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

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Find the points at which the tangent is horizontal.

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For Horizontal tangent: $\"\"$ .

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Consider $\"\"$ .

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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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$\"\"$ .

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Equate $\"\"$ to $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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Horizontal tangents are at $\"\"$ .

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$\"\"$

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Find the set of points.

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At $\"\"$ :

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$\"\"$

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$\"\"$ .

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At $\"\"$ , the point is $\"\"$ .

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At $\"\"$ :

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$\"\"$

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$\"\"$ .

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At $\"\"$ , the point is $\"\"$ .

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At $\"\"$ :

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$\"\"$

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$\"\"$ .

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At $\"\"$ , the point is $\"\"$ .

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Horizontal tangent line occur at points $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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For Vertical tangent: $\"\"$ .

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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$ .

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Equate $\"\"$ to $\"\"$ .

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$\"\"$ .

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Hence there are no solutions.

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No vertical tangents.

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$\"\"$

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Horizontal tangent line occur at points $\"\"$ , $\"\"$ and $\"\"$ .

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No Vertical tangents.