Problem 97AE

Sector of a hyperbola Let H be the right branch of the hyperbola x2 − y2 = 1 and let ℓ be the line y = m(x − 2) that passes through the point (2, 0) with slope m, where −∞<m<∞. Let R be the region in the first quadrant bounded by H and ℓ (see figure). Let A(m) be the area of R. Note that for some values of m, A(m) is not defined.

a. Find the x-coordinates of the intersection points between H and ℓ as functions of m; call them u(m) and υ(m) where υ(m) > u(m) > 1. For what values of m are there two intersection points?

b. Evaluate and .

c. Evaluate and .

d. Evaluate and interpret .

Solution 97AE

Step 1:

Given : and

a. To find the point of intersection of the hyperbola and line, we put in the equation of the hyperbola .

This represents a quadratic equation of form so the roots of the equation are :

Putting the value of , we get

and

Hence the x coordinate of the point of intersection of H and l are :

and

b.

.

Now,

Hence x coordinate of one of the points of intersection is but the other is undefined as m tends to 1.

c.

.

Now,

Hence x coordinate of the points of intersection is 2 for both points as m tends to .

d. Now we find the area bounded by hyperbola and the line as follows,

(Using =())

Now evaluating , we get