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 .
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
Hence the x coordinate of the point of intersection of H and l are :
Hence x coordinate of one of the points of intersection is but the other is undefined as m tends to 1.
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,
Now evaluating , we get