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 v(m), where v(m) > u(m) > 1. For what values of m are there two intersection points?

b. Evaluate

c. Evaluate

d. Evaluate and interpret 

Transcribed Image Text:

lim u(m) and lim v(m). т>1+ т>1+ lim u(m) and lim v(m). т—00 тЭ0