(a)

The hyperbola is .

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Let be the tangent point with coordinates .

Slope of the tangent line is the derivative of the curve.

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Differentiate on each side with respect to .

Find the slope at .

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Find the tangent line equation.

Point -slope form of line equation is .

Substitute and point in the above formula.

Substitute in the above expression to find intercept.

Substitute in the tangent line to find intercept.

Therefore line joining the points are and .

Find the mid point of the line joining points.

Thus the mid point of the line segment cut from the coordinate axes is .

(b)

Tangent line intersects axis at , axis at .

There formed a right angle triangle by the coordinate axes and tangent line.

Base of the triangle is and height of the triangle is .

Area of the right angle triangle with base and height is given by,

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Substitute and in the above formula.

Since area of the triangle does not contains coordinates of .

Therefore area of the triangle does not depends on position of .

(a) Mid point of the line segment cut from the coordinate axes is .

Area of the triangle does not depends on position of .