$\"\"$

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(a) Find the curvature of a straight line.

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The parametric line equations with slope $\"\"$ is $\"\"$ and $\"\"$ .

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Apply curvature formula: $\"\"$ .

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Find the first and second derivatives with respect to $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

\

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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The curvature at each point of a straight line is $\"\"$ .

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$\"\"$

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(b) Find the curvature of a circle.

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

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Apply curvature formula: $\"\"$ .

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Find the first and second derivatives with respect to $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

\

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

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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 curvature of any circle is $\"\"$ .

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$\"\"$

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(a) The curvature of any straight line is $\"\"$ .

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(b) The curvature of any circle is $\"\"$ .