$\"\"$

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The catenary $\"\"$ .

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Let the point $\"\"$ be a point on the catenary.

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consider $\"\"$ .

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Differentiate the function with respect to $\"\"$ .

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$\"\"$

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Therefore, the slope at point $\"\"$ is $\"\"$ .

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$\"\"$

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Since the slopes of the perpendecular lines negative reciprocals, consider the slope of $\"\"$ is $\"\"$ .

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The line $\"\"$ passes through the vertex $\"\"$ with the slope of $\"\"$ .

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The equation of the line $\"\"$ :

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$\"\"$

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When $\"\"$ , $\"\"$ .

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$\"\"$

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The length of $\"\"$ $\"\"$ .

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$\"\"$

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$\"\"$ , the ordinate $\"\"$ of the point $\"\"$ .

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Therefore, the lenght of $\"\"$ intercepted by the axes is equal to the ordinate $\"\"$ of the point $\"\"$ .

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$\"\"$

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The lenght of $\"\"$ intercepted by the axes is equal to the ordinate $\"\"$ of the point $\"\"$ .