$\"\"$

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Statement: The families of $\"\"$ and $\"\"$ are mutually orthogonal.

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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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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Slope of $\"\"$ is $\"\"$ .

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$\"\"$

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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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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Slope of $\"\"$ is $\"\"$ .

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The product of slopes of $\"\"$ and $\"\"$ :

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$\"\"$

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$\"\"$ .

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Since the product of slopes is $\"\"$ , $\"\"$ and $\"\"$ are orthogonal.

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The statement is true.

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$\"\"$

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The statement is true.