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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.1, Problem 70E

(a)

To determine

**To show:** The midpoint of the line segment cut from the tangent line by coordinate axes *P*.

Expert Solution

**Given:**

The equation of the hyperbola is

**Derivative rules:**

(1) Power Rule:

(2) Constant multiple rule:

**Formula Used:**

The equation of the tangent line at

where, *m* is the slope of the tangent line at

**Proof:**

Obtain the slope of the tangent line to the hyperbola at *P*.

The derivative of hyperbola *y* is

Apply the constant multiple rule (2) and the power rule (1),

Thus, the derivative of *y* is

Therefore, the slope of the tangent to the hyperbola is

Since the hyperbola is *P* is *P* is

Substitute *m* in equation (1),

Thus, the equation of tangent line at *P* is

Substitute 0 for *x* in *y-*intercept is

Substitute 0 for *y* in *x*-intercept is computed as follows.

Thus, the *x*-intercept of line

The midpoint of the line segment joining the points

Therefore, it can be concluded that the midpoint of the line segment cut from the tangent line by coordinate axes is *P*.

(b)

To determine

**To show:** The triangle formed by the tangent line and the coordinate axis always has the same area.

Expert Solution

**Proof:**

From part (a), the *x*-intercept of the tangent line is 2*a* and the *y*-intercept of the tangent line is

Here, the base *b* is *x*-intercept of the tangent line and the height *h* is *y*-intercept of the tangent line.

The area of the triangle bounded by axes and tangent line is computed as follows.

It is obvious that the area of the triangle is a constant and it is independent of *x* and *y*.

Therefore, it can be concluded that the triangle formed by the tangent line and the coordinate axes always has the same area.