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The parabolas are $\"\"$ and $\"\"$ .

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Let $\"\"$ be tangent line to the both the curves at $\"\"$ and $\"\"$ respectively.

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Slope of the tangent line is derivative of the curve.

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$\"\"$

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Find slope at $\"\"$ :

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$\"\"$ .

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Compare this slope with slope of the line $\"\"$ .

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$\"\"$

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$\"\"$

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Find slope at $\"\"$ :

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$\"\"$ .

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Compare this slope with slope of the line $\"\"$ .

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$\"\"$

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$\"\"$

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

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The tangent line and function are equal at a particular point.

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Hence equate the function and the tangent line equation.

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$\"\"$

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Substitute point $\"\"$ in above expressions.

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$\"\"$

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

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$\"\"$

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Substitute point $\"\"$ in above expressions.

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$\"\"$

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Solve the equations $\"\"$ and $\"\"$ .

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From $\"\"$ and $\"\"$ we have,

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

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Substitute $\"\"$ in equation $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in equation $\"\"$ .

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$\"\"$

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Equate the results in $\"\"$ and $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in common tangent line equation $\"\"$ .

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$\"\"$

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Therefore common tangent line to both the curves $\"\"$ and $\"\"$ is $\"\"$ .

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$\"\"$

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Graph:

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Graph of the curves $\"\"$ and $\"\"$ .

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$\"\"$

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Graph of the curves $\"\"$ and $\"\"$ is

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$\"\"$

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Common tangent line equation is $\"\"$ .