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The equation of the curve is $\"\"$ .

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Consider a point on the curve be $\"\"$ .

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At the point $\"\"$ the hyperbola equation is $\"\"$ .

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Slope of the tangent is derivative of the function at a particular point.

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Find the derivative of the curve.

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Consider $\"\"$ .

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

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$\"\"$

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Slope of the tangent line at $\"\"$ is $\"\"$ .

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Find the tangent line equation at $\"\"$ .

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Point-slope form of line equation: $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in point-slope formula.

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$\"\"$

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Equation of a tangent line is $\"\"$ .

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Equation of a tangent line is $\"\"$ .

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Find $\"\"$ -intercept by substituting $\"\"$ in tangent line equation.

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$\"\"$

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Thus, $\"\"$ -intercept is $\"\"$ .

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Find $\"\"$ -intercept by substituting $\"\"$ in the tangent line equation.

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$\"\"$

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Thus, $\"\"$ -intercept is $\"\"$ .

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Sum of the $\"\"$ -intercept and $\"\"$ -intercept is

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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Sum of the $\"\"$ -intercept and $\"\"$ -intercept is $\"\"$ .

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Sum of the $\"\"$ -intercept and $\"\"$ -intercept is $\"\"$ .